Introduction to Linear Algebra for Science and Engineering, 3rd edition By Daniel Norman Test Bank

Introduction to Linear Algebra for Science and Engineering, 3rd edition BY Daniel Norman

Email: richard@qwconsultancy.com

Exam Name___________________________________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

1) Let u = -4 , v = -7 . Find u + v .

-1

1)

-9

A) -3

B) -13

-8

C)

-8

-5 -16

D) -11 -10

2) Let u = 2 , v = 2 . Find u - v.

4

A) -7

B)

-2

3) Let u = 8 , v = 8

A)

2)

9

0 -5

C)

4 13

D) -2 -7

4 . Find v - u.

-1

3)

B) 12

0 -5

C)

7

-9 -4

D) -4 -9

4) Let u = -5 . Find 6u.

4)

-2

A)

30 -12

C) 30

B) -30 12

12

D) -30 -12

5) Let u = -3 . Find 7u.

5)

2

A) -21 14

B) -21

C)

-14

21 -14

D) -21 -14

6) Let u = -2 . Find -9u.

6)

-3

A)

18 -27

B) 18

C) -18

27

27

1

D) -18 -27

7) Let u = A)

5 , v = -2 . Find 2u + v.

-6 -3 6 -15

B)

7)

6 -9

C) -2 -5

D)

8) Let u = -3 and v = -2 . Display the vectors u, v, and u + v on the same axes.

5

2

A)

B)

C)

D)

2

8 -15

8)

9) Let u =

5 . Display the vector 2u using the given axes.

-4

A)

B)

C)

D)

10) Consider the points P(1,2,2), Q(-1, 0, 1), and R(3, 2, 1). Then A) PQ + RP = QR

B) PQ + PR = QR

C) PQ + PR = PR

D) QP + PR = QR

3

9)

10)

11) Find a vector equation of the line passing through the point P(1, -1, 3) and parallel to the line with 2 4 an equation X = 1 + t -2 , t 2 2

2

4

2

2

1

2

3

2

A) X = 1 + t -2 , t

C) X = -1 + t 1 , t

.

.

2

1

2

3

B) X = 1 + t -1 , t

.

1

2

3

1

.

D) X = -1 + t -1 , t

.

12) Find a vector equation of the line passing through the points P(-2, 1, 3) and Q(3, 0, -2).

-2

1

3

1

3

1

-2

1

A) X = 1 + t 1 , t

C) X = 0 + t 1 , t

11)

.

2

1

3

-5

-2

5

3

-5

B) X = 1 + t 1 , t

.

D) X = 1 + t -1 , t

12)

.

.

13) Find the parametric equations for the line passing through the points P(5,-1) and Q(2,3).

A) x1 = -3 + 5t B) x1 = 2 - 3t C) x1 = 5 + 2t D) x1 = 3 + 2t

13)

14) Find the parametric equations of the line passing through the point (1,-1,2) and parallel to the line

14)

x2 = 4 - t

x 2 = 3 + 4t

x 2 = -1 + 3t

x 2 = -4 + 3t

x 1 = 2 + 5t, x 2 = 3 - 2t, x 3 = -2 + t.

A) x1 = 1 + 5t, x 2 = 4 - 2t, x 3 = -4 + t C) x1 = 1 + 2t, x 2 = -1 + 3t, x 3 = 2 - 2t

B) x1 = 1 + 5t, x 2 = -1 - 2t, x 3 = 2 + t D) x1 = 7 + 5t, x 2 = 5 - 2t, x 3 = t 1

0

0

1

15) Determine which vector from the options below is in the plane: x = s -1 + t 2 .

0

1

A) 3 0

1

2

C) 8

B) 1 1

D) 3 3

-1

16) Determine a scalar equation for the line through the points P(3,1), Q(0,-2). (x 1 - 3)

A) x2 = x1 - 2

B) x2 = 1 -

x -3 C) x2 = 3 + 1

x D) x2 = -2 + 1

3

3

3

4

15)

16)

17) Let a1 =

4 -1 -14 1 , a2 = 3 , and b = 3 .

-2 3 12

17)

Determine whether b can be written as a linear combination of a1 and a2 . In other words, determine whether weights x 1 and x 2 exist, such that x 1 a1 + x 2 a2 = b. Determine the weights x 1 and x 2 if possible.

A) x1 = -2, x2 = 1

B) x1 = -3, x2 = 3

1

-3

2

-3

1

6

18) Let a1 = 2 , a2 = -4 , a3 = 1 , and b =

C) x1 = -3, x2 = 2

D) No solution

-4 2 .

2

18)

Determine whether b can be written as a linear combination of a1 , a2 and a3 . In other words, determine whether weights x 1 , x 2 , and x 3 exist, such that x 1 a1 + x 2 a2 + x 3 a3 = b. Determine the weights x 1 , x 2 , and x 3 , if possible.

A) x1 = -6, x2 = 0, x3 = 1

B) No solution

C) x1 = -2, x2 = -1, x3 = 2

D) x1 = 2, x2 = 1, x3 = - 3 2

19) Determine which of the following sets are linearly independent.

V=

1 2 -3 -3 , 8 , -2 . , U = -6 8 5

A) V only

-2 4 , 2

19)

1 4 0 , -4 .

-12 7

B) U only

C) V and U

20) For what values of h are the given vectors linearly independent?

1 -4 -6 , 24 1 h

A) Vectors are linearly dependent for all h.

B) Vectors are linearly independent for h = -4.

C) Vectors are linearly independent for h -4.

D) Vectors are linearly independent for all h.

5

D) None of them 20)

21) For what values of h are the given vectors linearly dependent?

21)

5 6 -1 -24 4 , 2 , 2 , -8 6 -3 6 h

A) Vectors are linearly dependent for h -24.

B) Vectors are linearly dependent for h = -24.

C) Vectors are linearly dependent for all h.

D) Vectors are linearly independent for all h.

22) Determine which of the following sets form a basis for R3 .

U=

1 -1 0 , 1 2 3

, V=

1 0 0 0 , 0 , 1 0 1 0

A) V only

,W=

1 0 2 1 2 , 8 , 1 , 2 -1 -1 0 1

B) U and V

22) and X =

C) V and X

1 0 -1 1 , -1 , 0 0 1 1

D) V and W

23) Determine which of the following sets form a basis for R3 .

U=

1 1 , 0 1

, V=

A) U and X

1 3 , 2 6

,W=

1 3 , 2 6

-1 1 , 5 -4

C) U only

D) V and X

?

24)

-1 1 0 4 ,v = ,w= ,x = -2 1 5 8

A) w and x

B) v and x

25) Which vectors are in the span of B = u=

,X=

B) U and V

24) Which vectors are in the span of B = u=

-3 2 , 6 -4

23)

1 0 , 2 2

C) w

D) x

?

25)

-1 1 0 4 ,v = ,w= ,x = -2 1 5 8

A) v

B) v , w and x

C) w and x

D) All of the vectors are in the span

6

26) Compute the dot product u · v where u = -15 , v = -14 .

9

A) 210

26)

-4

B) 174

C) -36

D) 246

27) Compute the dot product u · v where u = -12 , v = 0 .

4

A) 72

27)

15

B) 48

C) -180

D) 60

28) Compute the dot product u · v where u = 1 , v = 7 .

14

A) 23

B) 21

29) Compute the dot product u · v where u = A) 0

C) 100

A) -21

D) 7

5 -1 3 ,v = 2 .

3 -3

C) 8

B) -8

30) Compute the dot product u · v where u =

28)

1

29) D) -2

2 -16 = , v 0 3 .

-8 -1

B) -24

C) -40

31) Find the distance between the two points, P (6, -1), and Q (1, 6).

A) 37 B) 74 C) 74 32) Find the distance between the two points, P (-8, 16), and Q (16, -16).

A) 8 B) 1,600 C) 200 33) Find the distance between the two points, P (0, 0, 0), and Q (6, 9, 9).

A) 198 B) 3 22 C) 24 34) Find the distance between the two points, P (0, 0, 0), and Q (-8, -6, -2).

A) -16 B) 104 C) 2 -1 35) Find the distance between the two points, P (-8, 2, -6), and Q (-3, 3, 4).

A) 126 B) 16 C) 3 14

30) D) 0 31) D) 37 32) D) 40 33) D) 2 6 34) D) 2 26 35) D) 5 6

36) For a set of vectors to be orthogonal, each vector in the set must be orthogonal to every other vector in the set. Determine which of the following set of vectors are orthogonal.

V=

3 -3 3 6 , 0 , -3 3 3 3

A) Neither

,U=

20 -20 20 40 , 0 , 20 20 20 20

B) U and V

C) U only

7

D) V only

36)

1 -1 k , k is orthogonal.

1 -1 C) k = ± 2

37) Determine for what values of k the pair of vectors A) - 2

B) 0 1

0

1

1

37) D) 2

38) Find the angle between the vectors x = 0 and y = 1 .

A) /2

B) /3

38)

C) /4

D) /6

39) Find a scalar equation for the plane through the point P(-1, 2, 4) to the plane 2x + y - 2z = 4.

A) 2x + y - 2z = -4 B) 2x + y - 2z = -8 C) 2x + y - 2z = 4 D) 2x + y - 2z = 8

39)

40) Find an equation of the plane such that each point of the plane is equidistant from the points P(1,

40)

2, 1) and Q(-1, 0, 3).

A) x + y - z = -1

B) x + y - z = 3 3

1

2

1

C) x + 2y - z =- 1

D) -x + y - z = -1

41) Find the cross product: -2 × -2 .

2

41)

-4

1

1

0

D) 1

C) -1 -4

-4

0

2

-5

-2

B) 1

A) -1

-4

42) Find the cross product: 0 × 0 .

0

A) -1 0

42) 0

1

B) 0

0

C) 0

D) 1

1

0

0

43) Find the scalar equation of the plane that contains (1,1,1), (3,-1,5), and (4,2,-2).

A) 8x 1 + 2x 2 + 4x 3 = 14

B) x1 + 9x 2 + 4x3 = 14

C) -2x 1 + 9x2 + x 3 = 8

D) x1 + x 2 + 5 x3 = 9/2 2

8

43)

44) Determine a vector equation of the line of intersection of the planes x + y + z = 3 and x - y + z = 2.

1

2

-1

1

C) x = 0 + t -1 , t

1

2

-4

1

D) x = 0 + t -1 , t

1

2

0

0

45) Calculate the area of the parallelogram induced by u = 2 , v = 3 .

A) 7

B) 5

45) D) 2

C) 1

1 1 -1 0 +t 0 +s 2 .

2 1 -1 B) 4x + 3y - 2z = - 6

46) Determine the scalar equation of the plane with vector equation x = A) 4x + 3y - 2z = 6 C) -4x - 3y + 2z = -1

46)

D) -4x - 3y + 2z = 6

47) Determine which of the following sets is a subspace of R2 V is the line y = x in the xy-plane: V =

47)

x :y=x y

W is the union of the first and second quadrants in the xy-plane: W = U is the line y = x + 1 in the xy-plane: U =

A) V only

44)

-1 3 B) x = 0 + t 1 , t 4 -5

-1 5/2 A) x = 1/2 + t 0 , t 0 1

x :y 0 y

x :y=x+1 y

B) U and V

C) U only

D) W only

48) Let H be the set of all points of the form (s, s -1). Determine whether H is a vector space. If it is not a vector space, determine which of the following properties it fails to satisfy.

A: Contains zero vector B: Closed under vector addition C: Closed under multiplication by scalars

A) H is a vector space.

B) H is not a vector space; fails to satisfy all three properties.

C) H is not a vector space; does not contain zero vector.

D) H is not a vector space; not closed under vector addition.

9

48)

49) Let H be the set of all points in the xy-plane having at least one nonzero coordinate: H=

49)

x : x, y not both zero y

Determine whether H is a vector space. If it is not a vector space, determine which of the following properties it fails to satisfy: A: Contains zero vector B: Closed under vector addition C: Closed under multiplication by scalars

A) H is not a vector space; does not contain zero vector.

B) H is not a vector space; not closed under vector addition.

C) H is not a vector space; fails to satisfy all three properties.

D) H is not a vector space; does not contain zero vector and not closed under multiplication by scalars.

-1 0 1 50) Compute the linear combination: 3 1 + 2 - 4 .

10 5 -4 2 -4 -1 3 0 A) 1 B) 9 C) 3 -7 1 1

50) -2 1 -7 3

D)

0 -1 -7 3

51) Determine which of the following is a basis for the subspace spanned by the vectors in the set

51)

2 -1 2 , 0 0 1 2 -1

i)

1 0 0 , 1 0 1 1 0

1 0 1 , 1 0 1 1 0

ii)

A) iii only

iii)

1 0 -1 , 1 1 0 0 1

iv)

B) ii only

1 0 0 0 , 1 , 1 0 1 0 0 0 0

C) iii and iv only

D) ii and iv only

52) Determine which of the following forms a basis for the plane in R3 with scalar equation x 1 - 2x2 + x 3 = 0.

i)

2 -2 1 , 1 1 1

A) iii only

ii)

1 0 -2 , 0 1 0

iii)

2 -1 1 , 0 0 1

B) ii only

iv)

1 1 1 , 0 -1 1

C) iii and iv only

10

D) i and iv only

52)

53) Find the projection of v . u = A) 16 -8

8 ,v = 8 .

-24 -4

B) 16

C)

-4

54) Find the projection of u onto v . u = -3 , v = -4

A)

25 -50

53)

B)

D) 4

8 -4

2

5 .

-10

1 -2

54)

C)

1

1

1

2

1 5

5 -10

D)

2 5

-

55) Find the projection of u onto v . u = -2 and v = 1 .

1 4 1

1 6

55) 1 6

1 6

1

1

1

A) 4

B) - 6

C) 6

D) - 3

1 2

1 3

1 3

1 6

1 1 0 + t -1 , t 1 -1 B) 4 , 1 , - 8 3 3 3

56) Find the point on the line x = A) 4 , 1 , 2 3 3 3

, closest to Q(0, 0, 1).

C) 4 , - 1 , - 2 3

3

42 3

B)

5 3

D) 2 , - 1 , - 2

3

1 1 + t 0 -1 , t 1 -1 10 C) 3

57) Find the distance from the point Q(0,0,1) to the line x = A)

56) 3

3

..

57) D)

2 3

58) Find the distance from the point Q(1,-1, 2) to the plane x + y + z = 1.

A) 1 2

B)

2 3

C) 1 3

11

3

58) D) 1

6

59) Find a unit vector in the direction of 16 .

59)

-32

A)

1 5 2 5

B)

1 5

60) Find a unit vector in the direction of 2 9 2 9 1 9

D)

2 3 2 3 1 3 -

C)

D)

61) Determine a normal vector of the hyperplane 3x 1 + 2x 2 - 4x 3 = 5 in R4 .

A)

3 2 -4 5

B)

3 2 -4 1

62) Find the distance between the lines x = A) 3

2 5

60)

2 5 2 5 1 5

B)

2 3

1 5

-32 32 .

-16

-

-

A)

C)

2 5

1 3

C)

1 -1 0 +t 0 ,t 1 -2

3 2 4 0

2 3 2 3 1 3

61) D)

3 2 -4 0

1 1 , and x = -2 + s 1 s ..

1 0 C) 5 D) 1

B) 2

12

62)

Answer Key Testname: UNTITLED1

1) D 2) B 3) D 4) D 5) A 6) B 7) D 8) C 9) D 10) D 11) D 12) D 13) B 14) B 15) B 16) A 17) C 18) B 19) C 20) C 21) C 22) C 23) D 24) A 25) D 26) B 27) D 28) B 29) B 30) B 31) B 32) D 33) B 34) D 35) C 36) D 37) C 38) B 39) B 40) A 41) A 42) D

13

Answer Key Testname: UNTITLED1

43) A 44) A 45) C 46) A 47) A 48) B 49) C 50) C 51) B 52) C 53) A 54) B 55) C 56) C 57) A 58) D 59) A 60) C 61) D 62) A

14

Exam Name___________________________________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

1) Solve the system using back substitution.

1)

x 1 - x 2 + 3x 3 = -8 x2 + x 3 = 0 x3 = 0

A) (8, 8, 0)

B) (0, 8, -8)

C) (0, -8, -8)

D) (-8, 0, 0)

2) Solve the system using back substitution.

2)

x 1 - x 2 + 3x 3 = 6 x2 + x3 = 1

A) (8, 2, 0)

B) (0, 0, 2)

7

-4

0

1

C) x = 1 + t -1

D) No solution

3) The augmented matrix for a system is given as, 1 -5 -5 . Find the general solution in standard form or state that there is no solution.

0 0 7

A) x = -5 + t 5 y

0

1

x

-5

5

z

0

1

B) No solution

C) y = 7 + t 0

D) (-5, 7)

1

3)

1 2 -3 -9

4) The augmented matrix for a system is given as, 0 1 4 8 . Find the general solution in standard form or state that there is no solution.

x

A) y = z

x

C) y = z

0 0 0

6

3 -9 -2 0 +t 1 +s 0 0 0 1

B) No solution

11 -25 8 + t -4 0 1

D) y =

x z

11 -25 8 + t -4 6 0

1 2 -3 5

5) The augmented matrix for a system is given as, 0 1 4 -6 . Find the general solution in standard form or state that there is no solution.

x

5

z

0

A) y = 0 + t

-2 1 +s 0

3 0 1

x

17

11

z

0

1

B) y = -6 + t -4

D) x = 17

y = -6 - 4z z is free

y = -6 z=1

1 0

6

7

0 0

0

0

6) The augmented matrix for a system is given as, 0 1 -2 -2 . Find the general solution or state that there is no solution.

y=t

5)

0 0 0 0

C) x = 5 -2y + 3z

A) x = 7 - 6t

4)

B) No solution

C) x = 7 - 6t

y = -2 + 2t z=t

1 z=1+ y 2

2

D) x = 7 - 6t

y = -2 + 2t z=0

6)

1 4 -2 - 3 1

7) The augmented matrix for a system is given as, 0 0 1 3 5 . Find the general solution in 0 0 parametric form or state that there is no solution.

0

0 0

B) x1 = 11 - 4t - 3s

A) x1 = 11 - 4t - 3s x2 = t

x 2 = 5 - 3s

x 3 = 5 - 3s

x3 = s

x4 = s

C) x1 = 11 - 4t - 3s

D) x1 = - 4t + 2r + 3s + 1

x2 = t

x2 = 0

x 3 = 5 - 3s

x 3 = 5 - 3s

x4 = 0

x4 = s

1 6 6 -1 2 5

8) The augmented matrix for a system is given as, 0 0 0 -4 3 4 . Find the general solution in parametric form or state that there is no solution.

8)

0 0 0 0 -2 8

A) x1 = -6x2 - 6x3 + x4 - 2x5 + 5

B) x1 = -6x2 - 6x3 + 9

x2 is free

x2 is free

x3 is free

x3 is free

x4 =

7)

3 x -1 4 5

x4 = -4 x5 = -4

x5 = -4

C) No solution

D) x1 = -6x2 - 6x3 + 9 x2 is free x3 = -4 x4 =

3 x -1 4 5

x5 = -4

9) Determine which of the following matrices are in row echelon form.

9)

1 6 5 -7 1 4 5 -7 A = 0 1 -4 -1 , B = 0 1 -4 -5 0 0 1 3 0 3 1 4

A) A only

B) B only

C) Neither

3

D) Both A and B

10) Determine which of the following matrices are in row echelon form.

10)

1 4 5 -7 1 0 0 -7 A = 4 1 -4 7 , B = 7 1 0 5 0 4 1 4 0 4 1 7

A) Both A and B

B) A only

C) Neither

D) B only

11) Determine which of the following matrices are in row echelon form.

11)

1 -5 -5 3 1 3 7 -7 A = 0 1 -4 -3 , B = 0 0 2 -2 0 0 0 5 0 0 0 0 0 0 0 0

A) B only

B) Neither

C) A only

D) Both A and B

12) Row reduce the matrix to obtain a row equivalent matrix in the echelon form.

12)

1 4 -2 3 -3 -11 9 -5 -2 3 4 -3

1 4 -2 3 3 4 0 0 -33 -41

B) 0 1

1 4 -2 3 3 4 0 0 -11 -8

D) 0 1 3 4

1 4 -2 3 3 4 0 0 -33 0

A) 0 1

1

C) 0 1

4 -2 3

0 11

0 3

13) Row reduce the matrix to obtain a row equivalent matrix in the echelon form.

1 4 -5 1 2 5 -4 -1 -3 -9 9 2

1

A) 0 0 1

C) 0 0

0 1 0

2 4 2

1

4 -5 1 2 1 -2 1 0 0 0 1 4

0 0 0

0 14 0 -4 1 4

B) 0

4 -5 1 -2 0 0

0 -2 0 -4 1 4

D) 0 1 -2 0 -4

0

4

1 0

3 0 14

0 0

0 1

4

13)

14) Solve the system of linear equations using row reductions or show that it is inconsistent.

14)

x 1 - x 2 + 3x 3 = -8 2x 1 + x 3 = 0 x 1 + 5x 2 + x 3 = 40

A) (0, 8, 0)

B) (0, -8, 0)

C) (-8, 0, 0)

D) No solution

15) Solve the system of linear equations using row reductions or show that it is inconsistent.

15)

x 1 + 3x 2 + 2x 3 = 11 4x 4 + 9x 3 = -12 x 3 = -4

A) (1, 6, -4)

B) (-4, 1, 6)

C) (1, -4, 6)

D) (6, 1, -4)

16) Solve the system of linear equations using row reductions or show that it is inconsistent.

16)

x 1 - x 2 + 8x 3 = -107 6x 1 + x 3 = 17 3x 2 - 5x 3 = 89

A) (5, -8, -13)

B) (-5, -8, 13)

C) (5, 8, 13)

D) (5, 8, -13)

17) Solve the system of linear equations using row reductions or show that it is inconsistent.

17)

4x 1 - x 2 + 3x 3 = 12 2x 1 + 9x 3 = -5 x 1 + 4x 2 + 6x 3 = -32

A) (2, -7, 1)

B) (2, 7, -1)

C) (2, 7, 1)

D) (2, -7, -1)

18) Solve the system of linear equations using row reductions or show that it is inconsistent. x+y+z=6 x - z = -2 y + 3z = 11

A) (0, 1, 2)

B) (1, 2, 3)

C) (1, -2, 3)

5

D) No solution

18)

19) Solve the system of linear equations using row reductions or show that it is inconsistent.

19)

x+y+z=7 x - y + 2z = 7 5x + y + z = 11

A) (1, 4, 2)

B) (1, 2, 4)

C) (4, 1, 2)

D) (4, 2, 1)

20) Solve the system of linear equations using row reductions or show that it is inconsistent.

20)

x-y+z=8 x+y+z=6 x + y - z = -12

A) (2, -1, -9)

B) (-2, -1, 9)

C) (-2, -1, -9)

D) (-2, 1, 9)

21) Solve the system of linear equations using row reductions or show that it is inconsistent.

21)

5x + 2y + z = -11 2x - 3y - z = 17 7x + y + 2z = -4

A) (-3, 0, 4)

B) (3, 0, -4)

C) (0, -6, -1)

D) (0, -6, 1)

22) Solve the system of linear equations using row reductions or show that it is inconsistent.

22)

7x + 7y + z = 1 x - 8y + 8z = 8 9x + y + 9z = 9

A) (-1, 1, 1)

B) (0, 0, 1)

C) (0, 1, 0)

D) (1, -1, 1)

23) Solve the system of linear equations using row reductions or show that it is inconsistent.

23)

2x + y = 0 x - 3y + z = 0 3x + y + z = 0

A) (1, 0, 0)

B) (0, 1, 0)

C) No solution

D) (0, 0, 0)

24) Solve the system of linear equations using row reductions or show that it is inconsistent. x+y+z=7 x - y + 2z = 7 2x + 3z = 15

A) No solution

B) (1, 0, 1)

C) (1, 2, 4)

6

D) (1, -1, 1)

24)

25) Solve the system of linear equations using row reductions or show that it is inconsistent.

25)

x + 3y + 2z = 11 4y + 9z = -12 x + 7y + 11z = -11

A) No solution

B) (1, 2, 2)

C) (0, 0, 0)

D) (0, 1, 0)

26) Solve the system of linear equations using row reductions or show that it is inconsistent.

26)

5x + 2y + z = -11 2x - 3y - z = 17 7x - y = 12

A) (1, -5, 1)

B) (1, -2, 1)

C) No solution

D) (1, 0, 1)

27) Solve the system of linear equations using row reductions or show that it is inconsistent.

27)

3x2 + x4 = -7 x1 + x2 + 2x3 - x4 = 12 3x1 + x3 + 2x4 = 12 x1 + x2 + 5x3 = 26

A) (1, -2, 1, -1)

B) (3, -2, 5, -1)

C) (1, 5, 1, -1)

D) (1, 0, 1, -1)

28) Solve the system of linear equations using row reductions or show that it is inconsistent.

2x1 - 5x2 + 3x3 = -1 -2x1 + 6x2 - 5x3 = 6 -4x1 + 7x2 = -13 x1

A)

x1

12 7/2 x2 = 5 + t 2 x3 0 0

B) x2 = x3

x1

x1

C)

7/2 12 x2 = 2 +t 5 x3 1 0

D) x2 = x3

7

12 7/2 5 +t 2 0 1

-1 -1 5 +t 2 0 1

28)

29) Suppose an economy consists of three sectors: Energy (E), Manufacturing (M), and Agriculture

29)

(A).

Sector E sells 70% of its output to M and 30% to A. Sector M sells 30% of its output to E, 50% to A, and retains the rest.

Sector A sells 15% of its output to E, 30% to M, and retains the rest.

Denote the prices (dollar values) of the total annual outputs of the Energy, Manufacturing, and Agriculture sectors by pe, pm , and pa , respectively. If possible, find equilibrium prices that make each sector's income match its expenditures. Find the general solution as a vector, with pa free.

pe

0.465 pa

pe

A) pm = 0.593 pa

0.356 pa

B) pm = 0.686 pa

pa

pa

pa

pa

pe

0.308 pa

pe

0.607 pa

D) pm = 0.481 pa

C) pm = 0.716 pa pa

pa

pa

pa

30) The network in the figure shows the traffic flow (in vehicles per hour) over several one-way

streets in the downtown area of a certain city during a typical lunch time. Determine the general flow pattern for the network. In other words, find the general solution of the system of equations that describes the flow. In your general solution, let x4 be free.

A) x1 = 600 - x4

B) x1 = 600 + x5

C) x1 = 500 + x4

D) x1 = 600 - x4

x2 = 400 - x4

x2 = 400 - x5

x2 = 400 - x4

x2 = 400 + x4

x3 = 300 - x4

x3 = 300 - x5

x3 = 300 - x4

x3 = 300 - x4

x4 is free

x4 = 300

x4 is free

x4 is free

x5 = 300

x5 is free

x5 = 200

x5 = 300

8

30)

31) The table shows the amount (in g) of protein, carbohydrate, and fat supplied by one unit (100 g) of

31)

three different foods.

Food 1 Food 2 Protein 15 35 Carbohydrate 45 30 Fat 6 4

Food 3 25 50 1

Betty would like to prepare a meal using some combination of these three foods. She would like the meal to contain 15 g of protein, 25 g of carbohydrate, and 3 g of fat. How many units of each food should she use so that the meal will contain the desired amounts of protein, carbohydrate, and fat? Round to 3 decimal places.

A) 0.302 units of Food 1, 0.238 units of Food 2, 0.085 units of Food 3

B) 0.360 units of Food 1, 0.204 units of Food 2, 0.055 units of Food 3 C) 0.326 units of Food 1, 0.247 units of Food 2, 0.059 units of Food 3 D) 0.280 units of Food 1, 0.192 units of Food 2, 0.164 units of Food 3 32) Row reduce the matrix to determine the rank. What is the rank?

32)

1 -2 2 -4 2 -4 7 -4 -3 6 -6 12

A) 3

B) 2

C) 1

D) 4

33) Determine the rank of the matrix.

1 0 -3 0 0 1 -3 0 0 0 0 1 0 0 0 0

33)

4 2 1 0

A) 5

B) 2

C) 4

D) 3

34) Determine which of the following matrices are in RREF.

A) iv only

B) i and ii

C) ii and iii

9

34)

D) ii and iv

35) Row reduce the matrix to obtain a row equivalent matrix in the reduced echelon form.

1 4 -5 1 2 5 -4 -1 -3 -9 9 2

2 4 2

1

0

3 0 14

0

0

0 1

1

4 -5 1 -2 0 0

1

B) 0

A) 0 1 -2 0 -4

C) 0 0

35)

4

0

0 -2 0 -4 1 4

1

D) 0 0

0 1 0

0 0 0

4 -5 1 -2 0 0

0 14 0 -4 1 4 1 1 1

2 0 4

36) Solve the system A| b by row reducing the augmented matrix to RREF and then writing the

36)

solution in parametric form, where 1 1 2 A = 3 -1 1 0 2 -1

A) (-1,-1,2)

1 b= 0 5

C) - 1 , - 1 , 1

B) x = 1 - 3 t 4

y=

4

2

2

D) (1,2,-1)

3 5 - t 4 4

z=t

37) Solve the system A| b by row reducing the augmented matrix to RREF and then writing the solution in parametric form, where 1 1 -2 A = 2 -2 8 0 1 -3

A) (3,8,2)

3 b= 8 2

B) x = 1 + t

C) x = 1 - t

y = 2 - 3t z=t

y = 2 + 3t z=t

10

D) x = 1 - 2t y=t z=t

37)

38) Solve the system A| b by row reducing the augmented matrix to RREF and then writing the

38)

solution in parametric form, where 1 1 -2 A = 2 -2 8 0 1 -3

3 b= 8 .

2

B) x1 = 4 - t

A) x1 = 2 - r + t

C) x1 = 2 + r - t

x 2 = -t

x 2 = 2r - t x3 = r

x3 =

x4 = t

x 2 = -2r + t

3 1 + t 4 2

x3 = r x4 = t

x4 = t

D) x1 = 2 - t x 2 = 2t x3 = t

x4 = 0

39) Find the general solution of the simple homogeneous "system" below, which consists of a single

39)

linear equation. Give your answer as a linear combination of vectors. Let x2 and x3 be free variables.

-2x1 - 14x2 + 8x3 = 0

x1

x1

x1

A) x2 = -7 x2 - 4 x2 x3

x3

x3

x1

-7

4

x3

0

1

x1

7

(with x2 , x3 free)

B) x2 = x2 1 + x3 0 (with x2 , x3 free)

x3

0

-4 0 (with x2 , x3 free) 1

x1

-7

4

1

0

C) x2 = x2 1 + x3

D) x2 = x2 0 + x3 1 (with x2 , x3 free) x3

1 -1 -1

40) Solve the homogenous system A| 0 , A = -1 -8 -2 .

40)

3 12 2

A) x = 6

1 y=- t 3 z=t

B) x = 2 t

C) x = 0

3

y=0 z=0

1 y= t 3 z=t

11

D) x = 2t y = -t z=t

1 -2 1

41) Solve the homogenous system A| 0 , A = 3 -6 6 .

41)

-1 2 2

A) x = 0 y=0 z=0

B) x = 2t

D) x = 2t - r

C) x = 2t y=t z=r

y=t z=0

y=t z=r

42) Find the general solution of the homogeneous system below. Give your answer as a vector.

42)

x1 + 2x2 - 3x3 = 0 4x1 + 7x2 - 9x3 = 0 -x1 - 3x2 + 6x3 = 0 x1

A)

x1

-3 x2 = 3 x3 1 x1

C)

-3

B) x2 = x3 3

-3 x 2 = x3 3 x3 0

x3

1

x1

3

D) x2 = x3 -3 x3

1

43) Describe all solutions of Ax = b, where

43)

2 -5 3 -1 A = -2 6 -5 and b = 6 .

-4 7 0 -13 Describe the general solution in parametric vector form. x1 x1 12 7/2 7/2 12 A) x2 = 2 + x3 5 B) x2 = 5 + x3 2 x3 1 0 x3 0 1 x1

C)

x1

-1 -1 x2 = 5 + x3 2 x3 0 1

D) x2 = x3

12

12 7/2 5 + x3 2 0 0

b1 1 -3 2 44) Let A = -2 5 -1 and b = b2 .

3 -4 -6 b3

44)

Determine if the equation Ax = b is consistent for all possible b1 , b2 , b3 . If the equation is not consistent for all possible b1 , b2 , b3 , give a description of the set of all b for which the equation is consistent (i.e., a condition which must be satisfied by b1 , b2 , b3 ).

A) Equation is consistent for all b1, b2, b3 satisfying 2b1 + b2 = 0.

B) Equation is consistent for all b1, b2, b3 satisfying -3b1 + b3 = 0.

C) Equation is consistent for all b1, b2, b3 satisfying 7b1 + 5b2 + b3 = 0.

D) Equation is consistent for all possible b1 , b2 , b3 . b1 1 -3 2 45) Let A = -2 5 -1 and b = b2 .

3 -6 -3 b3

45)

Determine if the equation Ax = b is consistent for all possible b1 , b2 , b3 . If the equation is not consistent for all possible b1 , b2 , b3 , give a description of the set of all b for which the equation is consistent (i.e., a condition which must be satisfied by b1 , b2 , b3 ).

A) Equation is consistent for all b1, b2, b3 satisfying -3b1 + b3 = 0.

B) Equation is consistent for all b1, b2, b3 satisfying 3b1 + 3b2 + b3 = 0.

C) Equation is consistent for all possible b1 , b2 , b3 .

D) Equation is consistent for all b1, b2, b3 satisfying -b1 + b2 + b3 = 0.

46) Find the conditions on a,b such that the system ax + y = 1 2x + y = b is inconsistent.

A) a 2 and b 1

46)

B) a = 2 and b 1

C) a 2 and b = 1

47) Find the conditions on a such that the system

D) a = 2 and b = 1 47)

x + (a+1)y - z = 1 x - (a-1)y - z = 1 2x + 2y + (a-1)z = 1 has a unique solution.

A) a = -1

B) a 0 and a -1 D) a = 0 or a = 1

C) a = 0

13

48) Consider the set S=

5 Let p = -3 and q = 5 3

2 3 5 1 , -1 , 4 3 0 -1 4 -2 -2

.

48)

10 12 .

-6 -8

Determine whether p and/or q is in the span of set S. A) p Span S and q Span S. B) p Span S and q Span S.

C) p Span S and q Span S.

D) p Span S and q Span S.

49) Write y as a linear combination of the other vectors, v1 = 5 , v2 = 3 , v3 = 11 , y = 9 4 5 5 B) y = 5v1 + 3v2 - 7v3

A) y = v1 - 2v2 + 2v3 C) y = 4v1 - 2v2 - v3

49)

10

D) y = -5v1 + 4v2 + 2v3 1

0

1

6

2

-2

1

18

50) Write y as a linear combination of the other vectors, v1 = 1 , v2 = 4 , v3 = -5 , y = -38 .

A) y = -2v1 - 2v2 + 5v3 C) y = 2v1 - 5v2 + 4v3

50)

B) y = 4v1 + 2v2 - 5v3 D) y = 7v1 + 2v2 - 5v3

51) Find all values of h such that y will be in the subspace of 3 spanned by v1 , v2 , v3 1 3 7 -1 2 , v2 = 4 , v3 = 0 , and y = 7 .

-4 -8 0 h A) all h -14 B) h = -14

51)

if v1 =

C) h = -14 or 0 1 1 , 0 0

52) Find a homogeneous system that defines the set Span

A) x1 - x2 + 2x3 - 2x4 = 0 C) 2x1 - 2x2 + x3 -x4 = 0

1 1 2 , 1 1 1

B)

0 3 0 , 1 .

1 0 1 1

52)

B) 2x1 - x2 + x3 - 2x4 = 0 D) x1 - 2x2 + 2x3 - 2x4 = 0

53) Find a basis for the plane -3x1 + 2x2 + x3 = 0.

A)

D) h = -28

53)

1 1 2 , -1 1 -1

C)

14

1 1 2 , 1 -1 1

D)

1 -1 2 , 1 1 -1

54) Determine whether the given set is a basis for the given hyperplane. a)

1 0 3 1 , 0 , 1 0 1 0 0 1 1

b)

6 0 -2 1 , 3 , 0 -3 0 1 0 1 1

for x1 - x2 + 2x3 - 2x4 = 0.

for x1 + 2x3 = 0.

A) Neither a nor b C) a only 55) Let =

54)

1 2 -3 -3 , 8 , -2 8 5 -6

B) Both a and b D) b only and C =

4 -2 1 4 , 0 , -4 2 7 12

.

55)

Determine whether the given set is a basis for the given hyperplane.

A) Only C is linearly independent.

B) Both and C are linearly independent.

C) Both and C are linearly dependent.

D) Only is linearly independent.

56) For what values of h are the given vectors linearly independent?

56)

1 -4 -6 , 24 1 h

A) Vectors are linearly dependent for all h.

B) Vectors are linearly independent for all h.

C) Vectors are linearly independent for h = -4.

D) Vectors are linearly independent for h -4.

57) For what values of h are the given vectors linearly dependent?

5 6 -1 -24 4 , 2 , 2 , -8 6 -3 6 h

A) Vectors are linearly dependent for all h.

B) Vectors are linearly independent for all h.

C) Vectors are linearly dependent for h = -24.

D) Vectors are linearly dependent for h -24.

15

57)

58) Given the set of vectors

1 0 0 , 1 0 2

, decide which of the following statements is true:

58)

A: Set is linearly independent and spans R3 . Set is a basis for R3 .

B: Set is linearly independent but does not span R 3 . Set is not a basis for R3 .

C: Set spans 3 but is not linearly independent. Set is not a basis for R3 .

D: Set is not linearly independent and does not span R3 . Set is not a basis for R3 .

A) A

B) B

59) Given the set of vectors

C) C

1 0 0 0 0 , 1 , 0 , 1 0 0 1 1

D) D

, decide which of the following statements is true:

59)

A: Set is linearly independent and spans R3 . Set is a basis for R3 .

B: Set is linearly independent but does not span R 3 . Set is not a basis for R3 .

C: Set spans R3 but is not linearly independent. Set is not a basis for R 3 .

D: Set is not linearly independent and does not span R3 . Set is not a basis for R3 .

A) A

B) B

60) Consider B =

1 -3 2 -3 , 8 , -2 -4 4 -6

C) C and C =

1 4 -2 4 , 0 , -4 -14 4 -5

D) D .

60)

Determine whether set B or C or both sets B and C form a basis for R 3 .

A) neither of them is a basis for R3 .

B) only C is a basis for R3.

C) both B and C are bases for R3 .

-2

61) Let v1 = -2 , v2 = 2

D) only B is a basis for R3 .

6 -3 1 , v3 = -10 , and H = Span v1 , v2 , v3 .

-2 14

61)

Note that v3 = 3v1 - 4v2 . Which of the following sets form a basis for the subspace H, i.e., which sets form an efficient spanning set containing no unnecessary vectors?

A: v1 , v2 , v3 B: v1 , v2 C: v1 , v3 D: v2 , v3

A) B, C, and D

B) B only

C) B and C

16

D) A only

62) Find a basis for the solution space of the homogeneous system.

62)

4x - 2y + 10z = 0 2x - y + 5z = 0 -6x + 3y - 15z = 0

A)

-5 1 1 , 0 , 0 1

B)

1 0 0 0 , 1 , 0 0 0 1

C)

1/2 -5/2 1 , 0 , 0 1

D)

1/2 0 -5/2 , 1 , 0 1

63) Determine the augmented matrix of the system of linear equations, and determine the loop current indicated in the diagram:

A)

I1 3 I2 = 5 2 I3

B)

I1 3 I2 = 2 5 I3

C)

17

I1 2 I2 = 3 2 I3

D)

I1 3 I2 = 5 4 I3

63)

64) Determine the system of linear equations for the reaction forces and axial forces in members for the truss shown in the diagram. Assume that all triangles in this diagram are equilateral.

R1 R2

N1 N2

A) N3 N4 N5 N6 N7 R1 R2

N1 N2

C) N3 N4 N5 N6 N7

R1

3 3 -2 -2 1 = Fv -2 .

2 3 2 2 -1 -1

R2

N1 N2

B) N3 N4 N5 N6 N7 R1

3 3 -2 -2 1 = Fv 2 .

2 3 2 2 1 1

R2

N1 N2

D) N3 N4 N5 N6 N7

18

3 3 -2 -2 1 = Fv -2 .

2 3 2 2 1 1

3 3 -2 -2 1 = Fv -2 .

2 3 -2 -2 1 1

64)

65) Find the vectors b and c and the matrix A so that the problem is set up as a canonical linear

65)

programming problem. Do not solve the problem.

Maximize

2x1 + x2 + 5x3

subject to

x1 + 5x2 - 2x3 27 -2x2 + 6x3 20 x1 0, x2 0, x3 0

2

2

A) b = 1 , c = 27 , and A = 1 5 -2 5

20

0 -2

2

1 0 5 -2 -2 6

C) b = 27 , c = 1 , and A = 20

5

B) b = 27 , c = 1 , and A = 1 5 -2

6

20

0 -2 6 5 2 2 1 5 D) b = 27 , c = 1 , and A = 1 5 -2 20 5 0 -2 6

66) Solve the linear programming problem.

Maximize

6x1 + 7x2

subject to

2x1 + 3x2 12

66)

2x1 + x2 8 x1 0, x2 0

A) maximum = 39 when x1 = 3 and x2 = 3 C) maximum = 28 when x1 = 0 and x2 = 4

B) maximum = 38 when x1 = 4 and x2 = 2 D) maximum = 32 when x1 = 3 and x2 = 2

67) Alan wants to invest a total of $21,000 in mutual funds and a certificate of deposit (CD). He wants

to invest no more in mutual funds than half the amount he invests in the CD. His expected return on mutual funds is 10% and on the CD is 4%. How much money should Alan invest in each area in order to have the largest return on his investments? What is his maximum one-year return?

A) Maximum one-year return is $1,680 when he invests $14,000 in mutual funds and $7,000 in the CD. B) Maximum one-year return is $2,100 when he invests $21,000 in mutual funds and $0 in the CD. C) Maximum one-year return is $1,260 when he invests $7,000 in mutual funds and $14,000 in the CD. D) Maximum one-year return is $840 when he invests $0 in mutual funds and $21,000 in the CD.

19

67)

68) Solve the word problem by creating and solving a system of linear equations.

68)

A store makes two different types of smoothies by blending different fruit juices. Each bottle of Orange Daze smoothie contains 10 fluid ounces of orange juice, 4 fluid ounces of pineapple juice, and 2 fluid ounces of blueberry juice. Each bottle of Pineapple Blue smoothie contains 5 fluid ounces of orange juice, 6 fluid ounces of pineapple juice, and 4 fluid ounces of blueberry juice. The store has 500 fluid ounces of orange juice, 360 fluid ounces of pineapple juice, and 250 fluid ounces of blueberry juice available to put into its smoothies. The store makes a profit of $1.50 on each bottle of Orange Daze and $1 on each bottle of Pineapple Blue. How many bottles of each smoothie should the store make to maximize its profit? What is the maximum profit?

A) Maximum profit is $80 when the store makes 40 bottles of Orange Daze and 20 bottles of Pineapple Blue.

B) Maximum profit is $87.50 when the store makes 35 bottles of Orange Daze and 35 bottles of Pineapple Blue.

C) Maximum profit is $85 when the store makes 30 bottles of Orange Daze and 40 bottles of Pineapple Blue.

D) Maximum profit is $75 when the store makes 50 bottles of Orange Daze and 0 bottles of Pineapple Blue.

69) Solve the word problem by creating and solving a system of linear equations.

Daniel decides to feed his cat a combination of two foods: Max Cat and Mighty Cat. He wants his cat to receive four nutritional factors each month. The amounts of these factors (a, b, c, and d) contained in one bag of each food are shown in the chart, together with the total amounts needed. a b c d Max Cat 4 3 1 6 Mighty Cat 5 2 2 3 Needed 54 37 20 60 The costs per bag are $40 for Max Cat and $35 for Mighty Cat. How many bags of each food should be blended to meet the nutritional requirements at the lowest cost? What is the minimum cost?

A) Minimum cost = $500 when he blends 20 bags of Max Cat and 20 bags of Mighty Cat. 3 3

B) Minimum cost = $700 when he blends 0 bags of Max Cat and 20 bags of Mighty Cat. C) Minimum cost = $610 when he blends 3 bags of Max Cat and 14 bags of Mighty Cat. D) Minimum cost = $541.25 when he blends 17 bags of Max Cat and 23 bags of Mighty Cat. 2

4

20

69)

70) Solve the word problem by creating and solving a system of linear equations.

70)

A furniture company makes two different types of lamp stand. Each lamp stand A requires 20 minutes for sanding, 48 minutes for assembly, and 6 minutes for packaging. Each lamp stand B requires 9 minutes for sanding, 32 minutes for assembly, and 8 minutes for packaging. The total number of minutes available each day in each department are as follows: for sanding 3600 minutes, for assembly 9600 minutes, and for packaging 2000 minutes. The profit on each lamp stand A is $30 and the profit on each lamp stand B is $22. How many of each type of lamp stand should the company make per day to maximize their profit? What is the maximum profit?

A) Maximum profit is $5400 when they make 180 of lamp stand A and 0 of lamp stand B.

B) Maximum profit is $6380 when they make 66 of lamp stand A and 200 of lamp stand B. C) Maximum profit is $6850 when they make 100 of lamp stand A and 175 of lamp stand B. D) Maximum profit is $6164 when they make 138 of lamp stand A and 92 of lamp stand B. 71) Solve the word problem by creating and solving a system of linear equations.

71)

The Acme Class Ring Company designs and sells two types of rings: the VIP and the SST. They can produce up to 24 rings each day using up to 60 total man-hours of labor. It takes 3 man-hours to make one VIP ring and 2 man-hours to make one SST ring. How many of each type of ring should be made daily to maximize the company's profit, if the profit on a VIP ring is $40 and on an SST ring is $35?

A) 16 VIP and 8 SST B) 12 VIP and 12 SST

C) 18 VIP and 6 SST

D) 14 VIP and 10 SST

72) Solve the word problem by creating and solving a system of linear equations.

72)

An airline with two types of airplanes, P1 and P2 , has contracted with a tour group to provide transportation for a minimum of 400 first class, 750 tourist class, and 1500 economy class passengers. For a certain trip, airplane P1 costs $10,000 to operate and can accommodate 20 first class, 50 tourist class, and 110 economy class passengers. Airplane P2 costs $8500 to operate and can accommodate 18 first class, 30 tourist class, and 44 economy class passengers. How many of each type of airplane should be used in order to minimize the operating cost?

A) 9 P1 planes and 0 P2 planes B) 9 P1 planes and 13 P2 planes

C) 11 P1 planes and 7 P2 planes

D) 7 P1 planes and 11 P2 planes

73) A summer camp wants to hire counselors and aides to fill its staffing needs at minimum cost. The

average monthly salary of a counselor is $2400 and the average monthly salary of an aide is $1100.

The camp can accommodate up to 45 staff members and needs at least 30 to run properly. They must have at least 10 aides, and may have up to 3 aides for every 2 counselors. How many counselors and how many aides should the camp hire to minimize cost?

A) 27 counselors and 18 aides B) 35 counselors and 10 aides

C) 18 counselors and 12 aides

D) 12 counselors and 18 aides

21

73)

Answer Key Testname: UNTITLED2

1) D 2) C 3) B 4) B 5) B 6) C 7) A 8) B 9) A 10) C 11) D 12) A 13) B 14) A 15) A 16) D 17) D 18) B 19) B 20) B 21) D 22) B 23) D 24) A 25) A 26) C 27) B 28) B 29) B 30) D 31) C 32) B 33) D 34) D 35) D 36) D 37) C 38) A 39) B 40) B 41) B 42) B

22

Answer Key Testname: UNTITLED2

43) B 44) D 45) B 46) B 47) B 48) A 49) D 50) C 51) B 52) A 53) C 54) B 55) D 56) D 57) A 58) B 59) C 60) D 61) A 62) C 63) A 64) B 65) B 66) D 67) C 68) C 69) D 70) B 71) B 72) B 73) D

23

Exam Name___________________________________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

1) Let A = -3 1 . Find 2A.

1)

0 2

A) -6 2 0 2

B) -1 3

C) -6 1

D) -6 2

0 2

2 4

2) Let B = -1 4 7 -3 . Find -4B. A) 4 2 5 7 C) -3 2 5 -5

0 4

2) B) 4 -16 -28 12 D) 4 4 7 -3

2

3) Let C = -2 and D = [1 2 3]. Find 1 CD.

A) [6 12 18]

3)

2

12

6

B) [17]

1

C) -6

2

36

6 12 18

2

4) Let C = -2 and D = [1 2 3]. Find 1 DC. 12

6

A) -6 36

1

B) [6 12 18]

2 6

0 4 . Find 4A + B. -1 6

A) 4 28

B) 4 16

4 48

4)

2

5) A = 1 3 and B =

3

D) -1 -2 -3

2

3

C) -1 -2 -3 6 12 18

D) [17]

5) C) 4 16

7 30

1 12

1

D) 4 7

7 12

6) Let A = 1 3 and B = 2 6

A) 1 5

5 30

1

7) Let C = -3 and D = 2

5

A) -6

0 4 . Find A2 - B. -1 6

6) C) 7 17

B) 7 17

15 36

15 12

D) 1

5 3 30

-1 3 . Find C - 4D. -2

B)

4

7) 5

-3 9 -6

C) -15 10

8) Let A = -1 2 and B = 1 0 . Find 3A + 4B. A) 2 2 B) -3 4 -1 7

5 6

49

-5 7

D)

-5 15 -10

8) C) -1 4

D) 1 6

9) Let A = -5 1 and B = -3 -9 . Find A + B. 4 13 8 1 -1 -16

B) -8 -8

10) Let A = -2 7 and B =

2 6 . Find A - B. -2 9

A)

-2 -8

A) 0 1

0 -17

9) 4 1

4 13

C) -8 -8

-1 16

-1 16

-6 1

D) -2 -2 9 0

10)

B) 4 1

C) -4 1

-4 1

0 -17

D) 0 -1 -4 17

11) Let A = -10 3 and B = 0 0 . Find A + B. 7 -9

A) Undefined

11)

0 0

B) 0 0

C) 10 -3

0 0

-7 9

2

D) -10 3 7 -9

4 6

12) Find the transpose of the matrix. -6 0 A) 6

0 5 4 -6 -5

B)

12)

-5 5 6 4 0 -6 5 -5

-5 5

C) 4 -6 -5 6

0

D) -6 0

5

4 6

13) Find the transpose of the matrix. 9

8 9 8 0 -7 0 -7

A)

8 9 8 9 -7 0 -7 0

C)

9 0 8 -7 9 0 8 -7

13)

B) 0 -7 0 -7

D)

9

8 9

0 -7 0 -7

9 8 9 8

8

14) Given A = -1 3 and B = -2 0 . Find the matrix product AB. 22

A)

2 -6 -1 5

14)

-1 4

C) 12 -1

B) -1 12 -6 8

8 -6

D)

20 -2 8

15) Given A = 0 -3 and B = -2 0 . Find the matrix product AB. 3

3

A) -3 3 -3 -9

15)

-1 1

B) -6 -6 3

C)

6

3 -3 -9 3

D)

16) Given A = -1 3 and B = 0 -2 7 . Find the matrix product AB, if it is defined.

16

A)

1 -3 2

3 6 -7 -20 -1 19

B)

0 -6 21 1 -18 12

D) 3 -7 -1

C) AB is undefined.

6 -20 19

3

06 -3 3

16)

17) Given A = 3 -2 1 and B = 0

4 -2

4 0 . Find the matrix product AB, if it is defined.

-2 3

A) AB is undefined.

B) 12 -8 4

C) 12 0

D) -8 16

17)

-6 16 -8 12 -6

0 12

4 -8

18) Given A = 0 -2 and B = -1 3 2 . Find the matrix product AB, if it is defined.

2

3

A) AB is undefined.

C)

18)

0 -3 1

B)

0 -2 6 -3 -2 7

D) 0 -6 -12

0 6 -2 -2 -3 7

0 -9

3

30

19) Given A = 1 3 -1 and B = -1 1 . Find the matrix product (AB)T, if it is defined.

2 0

5

05

A) 6 25

B)

6 0 25 -2

C) (AB)T is undefined.

D)

0 6 -2 25

0 -2

20) Compute the product or state that it is undefined. [-6 2 5] A) [174]

B) [-51]

6 0 -3

C) Undefined

6 -7 38 1 8 -8 1 -1 C) 3 8 1 6 -7 8

20) D)

-36 0 -15

21) Compute the product or state that it is undefined. -8 1 -1 A) [-63 -30]

19)

B) Undefined

4

21) D) -63 -30

22) Compute the product or state that it is undefined.

8 -5 2 8 -6 6

-1 1

C)

-13 6 12

-8 -5 8 6 6

A) Undefined

B) -2

8 -2

23) Compute the product or state that it is undefined. 5 -2 4 -5

A)

-24 -18 -4

22) D)

-4 -6 4

23) 8 12 16 -20

-32

C) Undefined

B) -46 0

-8 2 5 8 -6 6

D) -30

24) Write the following system in the form A x = b , which is as a matrix equation involving the

24)

product of a matrix and a vector on the left side and a vector on the right side.

4x1 + x2 - 2x3 = 5 2x1 - 3x2

A)

C)

=1

x1 4 1 -2 x = 5 2 2 -3 0 1 x3 x1 4 1 -2 x = 5 2 2 3 1 1 x3

B) x1 x2 x3 2 -3

D)

4 1 -2

2 -3 0

0

4 5 1 = 1 -2 x1 x2

= 5 1

25) The sizes of two matrices A and B are given. Find the sizes of the product AB and the product BA,

25)

if the products are defined.

A is 4 × 4, B is 4 × 4.

A) AB is 8 × 4, BA is 8 × 4.

C) AB is 4 × 4, BA is 4 × 4.

B) AB is 1 × 1, BA is 1 × 1.

D) AB is 4 × 8, BA is 4 × 8.

26) The sizes of two matrices A and B are given. Find the sizes of the product AB and the product BA, if the products are defined.

A is 2 × 1, B is 1 × 1.

A) AB is 2 × 1, BA is 1 × 1.

C) AB is 2 × 1, BA is undefined.

B) AB is 1 × 2, BA is 1 × 1.

D) AB is undefined, BA is 1 × 2.

5

26)

27) The sizes of two matrices A and B are given. Find the sizes of the product AB and the product BA,

27)

if the products are defined.

A is 2 × 3, B is 3 × 2.

A) AB is 3 × 3, BA is 2 × 2.

C) AB is undefined, BA is 3 × 3.

B) AB is 2 × 2, BA is undefined.

D) AB is 2 × 2, BA is 3 × 3.

28) The sizes of two matrices A and B are given. Find the sizes of the product AB and the product BA,

28)

if the products are defined.

A is 2 × 1, B is 2 × 1.

A) AB is 2 × 1, BA is 1 × 2.

C) AB is 2 × 2, BA is 1 × 1.

B) AB is undefined, BA is undefined.

D) AB is 1 × 2, BA is 2 × 1.

29) If A = 5 3 and B = -4 12 , for what value of k does AB = BA? 2 1

29)

8 k

A) k = 20 C) k = 0

B) k = -20 D) no value of k will result in AB = BA

30) Identify the indicated sub matrix.

30)

0 1 -7 -5 If A = 7 -1 0 2 , then find A12.

2 5 -2 0

A) 7

C) -5

B) 2 5 -2

2

D) 1

31) Identify the indicated sub matrix.

1

A) -1

6 -6 If A = 0 7

31)

4 1 0 -1 , then find A .

21 7 -4 4 7

C) 4

B) -6

-4

6

D)

7 4

4

0

1

5

3

7

32) Find the matrix product AB for the partitioned matrices. A = 2 -1 -3 , B = 0 32 20 8 -7 18 46 31

-4 -1 0

-8

-2 0 8 5 1 6 2 2 4 -1 0 3

32)

3

B) -12 -3 0 -9

A) -5 -6 14

28 -7 0 21

-4 -1 32 23

-4 -1 32 23

C) -17 -3 14 -1

D) -17 -3 14 -1

21 11 46 52

21 11 46 52

33) Given that I is an identity matrix and the product AB is defined, find the matrix product AB for

33)

the partitioned matrices.

A= 0 I ,B= W X I F Y Z

A) X W + XF

B) 0

Z FY FZ

C) Y

D) Y

Z Y + ZF

Z W + FY X + FZ

Z W + YF X + ZF

34) Determine whether A, C or both A and C are in the span of B. Consider

=

10 , 01 , 00 00 10 01

, A = 1 2 , and C = 3 2 .

2 -1 1 1

A) Neither A nor C is in the Span B. C) Both A and C are in the Span B.

B) Only A is in the Span .

D) Only C is in the Span .

35) Determine which of the sets of matrices is linearly independent.

A=

1 2 , 32 , 00 2 -1 1 1 1 1

,B=

34)

35)

10 , 01 , 00 00 10 01

A) Neither A nor B C) Both A and B

B) A only D) B only

36) Let A = -1 0 -1 and fA be the corresponding matrix mapping.

3 4 -2 Determine the domain and codomain of f A. A) Domain = R3 , Codomain = R3 .

B) Domain = R2 , Codomain = R3 .

D) Domain = R2 , Codomain = R2 .

C) Domain = R3 , Codomain = R2 .

7

36)

37) Let : R3 A)

R3 be a linear mapping defined by (x, y, z) = (5x, 4y, z).

104 105 105 B) 0 1 4 C) 0 1 4 015 001 001 000

2

Define a transformation T: R3

38)

R2 by T u = A x .

Find T u , the image of u under the transformation T.

A) -8 16 3 24

39) Let T: R2

B) 11

16

D) 48

C) 24

26

8 -6

D)

8 2 .

-3

38) Let A = -1 8 -1 and u = 3 4

37)

18

-3

R 2 be a linear transformation that maps u = -3 into -13 and maps v = 4 into 4 6 6

6 .

-8 Use the fact that T is linear to find the image of 3u + v.

A) -5 B) -21 C) 18 -6

-33 10

-2

1

0

0

1

0

0

0

0

1

0

0

1

40)

Suppose that T is a linear transformation from R3 into R2 such that T( e1 ) =

6 , T( e ) = 2 -3

2 , and T( e ) = -2 .

3 0 1 x1

Find a formula for the image of an arbitrary x = x2 in R 3 . x3

x1

6x1 + 2x2 - 2x3

A) T x2 = 2x1

B) T x2 =

x3

-3x1 + x3

x3

x1

6x1 - 3x2

x1

C) T x2 = 2x1

D) T x2 =

x3

x3

2x2 + x3

8

39)

D) -7

40) The columns of I3 = 0 1 0 are e1 = 0 , e2 = 1 , and e3 = 0 .

x1

500 040 001

6x1 + 2x2 - 2x3 -3x1

6x1 - 3x2 2x1

+ x3

41) Determine if either of the following mappings are linear.

41)

f(x 1 , x 2 ) = (x 1 + 2, x 2 ) and g(x 1 , x 2 ) = (|x 1 |, x 2 ).

A) g only

B) f only

C) Neither

42) Find the matrix of the linear transformation T: V

D) Both

W relative to B and C.

42)

Suppose B = {b1 , b2 } is a basis for V, and C = {c1 , c2 , c3 } is a basis for W. Let T be defined by T(b1 ) = -5c1 - 6c2 + 5c3 T(b2 ) = -5c1 - 12c2 + 2c3

A) -5 -6 5 0

6 3

-5

0

-6 5 -5 -12 2

C) -5

B) -6 -6 5 -3

-5

-5

5

2

D) -6 -12

43) Suppose B = {b1 , b2 , b3 } is a basis for V, and C = {c1 , c2} is a basis for W. Let T be defined by

43)

T(b1 ) = 5c1 + c2 T(b2 ) = 6c1 - 6c2 T(b3 ) = 5c1 - 6c2

5

1

A) 6 -6

5

6 0 5 -1

B) 5

6 5 1 -6 -6

5 -6

C) 6

D) 6

0 -1 1 -6 -6

44) Find the standard matrix of the following linear mapping.

44)

A Reflection about the x-axis in R3 .

A)

-1 0 0 0 1 0 0 0 1

1 0 0

B) 0 -1 0

C)

0 0 1

-1 0 0 0 -1 0 0 0 1

1 0 0

D) 0 1 0 0 0 1

45) Find the 4 x 4 standard matrix of the linear mapping described as a translation by the vector (4, -7, -9).

A)

1 0 0 4 0 1 0 -7 0 0 1 -9 0 0 0 1

B)

1 0 0 -4 0 1 0 7 0 0 1 9 00 0 1

C)

9

4 0 0 0 0 -7 0 0 0 0 -9 0 0 0 0 1

D)

0 0 0 4 0 0 0 -7 0 0 0 -9 0 0 0 1

45)

105

46) Suppose that S and T are linear mappings with matrices [S] = 0 1 4 and [T] = 001

Determine the matrix that represents S T, if it is defined.

-4 -11

A) -2 -10

B)

-1 -3

1 4 2 10 -1 -3

-4 -11 -1 -3

105 001

Determine the matrix that represents T S, if it is defined.

-4 -11

B)

-1 -3

46)

D) undefined

C) -2 10

47) Suppose that S and T are linear mappings with matrices [S] = 0 1 4 and [T] =

A) -2 10

1 4 2 2 .

-1 -3

-4 -11 2 10 -1 -3

C)

1 4 2 10 -1 -3

1 4 2 2 .

-1 -3

47)

D) undefined

48) Let v = 1 . Determine the matrix proj v .

2 1 2 5 5

A)

-

2 5

1 5

48)

2 5

2 5

B) 2 4

4 5

5

4 5

1 5

C) 1 4

5

5

D) 2 1

5

49) Find the standard matrix of the linear transformation T where T: R2

5

C)

2 2 2 2

1 1 -1 1

B)

-

2 2 2 2

-

2 2

D)

2 2

10

-

-

5

R 2 rotates points (about the

7 origin) through radians (with counterclockwise rotation for a positive angle).

4

A)

2 5

2 2

2 2

3 3

3 3

3 3

3 3

49)

50) Find the 4 x 4 standard matrix of the linear mapping. Rotation about the y-axis through an angle

50)

of 60°.

A)

0.5 0 - 3/2 0

0 1 0 0

3/2 0 0.5 0

0 0 0 1

C)

0.5 - 3/2 0 0

3/2 0.5 0 0

0 0 1 0

0 0 0 1

B)

1 0 0 0

0 0.5 3/2 0

0 3/2 0.5 0

0 0 0 1

D)

3/2 0 -0.5 0

0 1 0 0

0.5 0 3/2 0

0 0 0 1

51) Describe geometrically the effect of the transformation T. 0 Let A = 0 0

0 1 0

51)

0 0 , define a transformation T by T(x) = Ax. 1

A) Projection onto the x2 -axis C) Rotation

B) Projection onto the x2 x3-plane D) Reflection

52) Find the 3 × 3 standard matrix of the linear mapping.

52)

Rotate points through 45° and then scale the x-coordinate by 0.2 and the y-coordinate by 0.4.

A) 0.1 2

0.1 0

-0.2 2 0.2 0

0 0 1

0.1 2

0.1 2

0

0

0

1

0.1 2

B) 0.2 2 0

0 -0.2 0 0 0

D) 0.4

C) -0.2 2 0.2 2 0

-0.1 2 0.2 2 0

0 0 1

0 0 1

53) Determine the 3 × 3 standard matrix of the composition of mappings.

53)

Translate by x coordinate by 9 and y coordinate by 7 and then reflect through the line y = x.

A)

-1 0 -9 0 -1 -7 0 0 1

0 1 9

0 7 1

C) 9 0 0

B) 1 0 7

0 0 1

0 0 1

11

0 1 7

D) 1 0 9

0 0 1

54) Consider a reflection refl n : R3

3 R3 over the plane with normal vector n = 0 .

4

54)

Determine the standard matrix refl n .

A)

19/25 0 -8/25

0 1 0

-24/25 0 -7/25

B)

-5/25 0 -2/25

0 1 0

C)

-6/25 0 -8/25

0 0 0

-24/25 0 -32/25

D)

19/25 0 -24/25

0 -8/25 1 0 0 -32/25

-24/25 0 1

1 -3 0 18 0 2 and b = -10 .

1 -5 4 32

55) Let A = -4

55)

Define a transformation T: R3 R3 by T(x) = Ax. If possible, find a vector x whose image under T is b. Otherwise; state that b is not in the range of the transformation T.

A) b is not in the range of the transformation T. 3

B) -5 1

C)

3 1 -5 3

D) -5

0

12

1 -3

2

3

2 -5

3

-2

56) Let A = -3 4 -1 and b = -5 .

56)

Define a transformation T: R3 R3 by T(x) = Ax. If possible, find a vector x whose image under T is b. Otherwise; state that b is not in the range of the transformation T.

A) B)

2 0 -2 -5 -5 0

C) b is not in the range of the transformation T. 6

D) 3 3

57) Determine if the vector u is in the column space of matrix A and whether it is in the nullspace of

57)

A.

-4 1 -2 3 u= ,A= 3 -4 -2 -1

0 -1 -3 3

3 -4 0 6

A) Not in Col A, not in Nul A C) In Col A, not in Nul A

B) Not in Col A, in Nul A D) In Col A and in Nul A

58) Determine which of the following vectors belongs to the null space of the matrix A.

58)

2 -1 A = -2 3 -5 , u = 3 , and v = -1 .

-2 -2 10 1 1

A) v only

B) u and v

C) u only

D) None of them

1 0 -5 -2

59) Find a basis for the nullspace of the linear mapping fA, if A = 0 1 7 -4 .

59)

0 0 0 0

A)

-5 -2 7 , -4 1 0 0 1

B)

5 2 -7 , 4 1 0 0 1

C)

13

1 0 0 , 1 0 0

D)

1 0 0 , 1 7 -5 -2 -4

1 0 -5 0 -2 60) Find a basis for the nullspace of the linear mapping fA, if A = 0 1 3 0 3 .

0 0 0 1 1 0 0 0 0 0

A)

5 2 -3 -3 1 , 0 0 -1 0 1

B)

1 0 0 0 , 1 , 0 0 0 1 0 0 0

C)

1 0 0 1 -5 , 3 0 0 -2 3

60)

D)

-5 -2 3 3 1 , 0 0 -1 0 1

61) Find a basis for the nullspace of the linear mapping fA, if A = 1 -2 -5 -3 .

0

A)

5 3 -1 , -2 1 0 0 1

B)

2 3 -1 1 , -1 , -2 0 1 0 0 0 1

C)

1

1

61)

2

3 -1 -1 , -2 1 0 0 1

D)

2 5 3 1 , -1 , -2 0 1 0 0 0 1

1 -2 3 -3 -1

62) Find a basis for the nullspace of the linear mapping fA, if A = -2 5 -5 4 2 .

A)

1 0 0 1 5 , 1 -7 -2 -1 0

C)

-5 7 1 -1 2 0 1 , 0 , 0 0 1 0 0 0 1

B)

D)

62)

-1 3 -2 1 1 2 -3 3 -1 1 -1 2 0 0 , 1 , 0 , 0 0 0 1 0 0 0 0 1 7 0 -5 -1 -2 0 1 , 0 , 0 0 1 0 0 0 1

1 -2 2 -3

63) Find a basis for the column space of the matrix, if B = 2 -4 9 -2 .

23 5

A)

2 1 , 0 0 4 0 5

63)

-3 6 -6 9

B)

1 -2 2 , -4 6 -3

C)

1

14

1 0 0 , 1 0 0

D)

1 2 2 , 9 -3 -6

1 0 -5 0 -5 64) Find a basis for the column space of the matrix, if B = 0 1 5 0 4 .

0 0 0 1 1 0 0 0 0 0

A)

1 0 0 0 , 1 , 0 0 0 1 0 0 0

C)

5 5 -5 -4 1 , 0 0 -1 0 1

B)

1 0 -5 0 , 1 , 5 0 0 0 0 0 0

D)

1 0 0 , 1 0 0 0 0

65) Assume that the matrix A is row equivalent to B. Find a basis for the row space of the matrix A. 1 3 A= 2 4 1 -5 -3 -1

-4 -5 0 8

0 5 -3 3

1 -2 , B = 2 -4

1 0 0 0

3 -2 0 0

-4 3 -8 0

0 5 -23 0

64)

65)

1 -4 17 0

A) {(1, 0, 0, 0), (3, -2, 0, 0), (-4, 3, -8, 0)} B) {(1, 3, -4, 0, 1), (2, 4, -5, 5, -2), (1, -5, 0, -3, 2), (-3, -1, 8, 3, -4)} C) {(1, 3, -4, 0, 1), (0, -2, 3, 5, -4), (0, 0, -8, -23, 17), (0, 0, 0, 0, 0)} D) {(1, 3, -4, 0, 1), (0, -2, 3, 5, -4), (0, 0, -8, -23, 17)} 66) Assume that the matrix A is row equivalent to B. Find a basis for the column space of the matrix

66)

A.

1 3 -4 0 1 2 4 -5 5 -2 , B = A= 1 -5 0 -3 2 -3 -1 8 3 -4

1 3 -4 0 0 -2 3 5 0 0 -8 -23 0 0 0 0

1 -4 , 17 0

A) {(1, 2, -1, -3), (3, 4, -5, -1), (-4, -5, 0, 8)} B) {(1, 0, 0, 0), (3, -2, 0, 0), (-4, 3, -8, 0)} C) {(1, 2, -1, -3), (3, 4, -5, -1), (-4, -5, 0, 8), (0, 5, -3, 3)} D) {(1, 3, -4, 0, 1), (0, -2, 3, 5, -4), (0, 0, -8, -23, 17)} 67) If the nullspace of a 7 × 5 matrix is 2-dimensional, find Rank A, Dim Row A, and Dim Col A. A) Rank A = 5, Dim Row A = 5, Dim Col A = 3 B) Rank A = 3, Dim Row A = 3, Dim Col A = 3 C) Rank A = 3, Dim Row A = 2, Dim Col A = 2 D) Rank A = 3, Dim Row A = 3, Dim Col A = 2

15

67)

68) If A is a 5 × 9 matrix, what is the smallest possible dimension of Nul A? A) 4 B) 5 C) 9 1 0 -5 0 -2 69) Find the rank of the matrix A = 0 1 3 0 3 .

0 0 0 1 1 0 0 0 0 0 A) 5 B) 3

68) D) 0

69) C) 4

D) 2

70) Find a basis for the row space, a basis for the column space, and a basis for the nullspace for A=

1 1 3 1 6 2 -1 0 1 -1 .

-3 2 1 -2 1

A) A basis for the row space is

1 0 1 0 -1 , 0 1 2 0 3 , 0 0 0 1 4

A basis for the column space is

A basis for the nullspace is

B) A basis for the row space is

-1 1 -2 -3 1 , 0 0 -4 0 0

C) A basis for the row space is

.

1 0 0 , 0 0 1

-1 1 -2 -3 1 , 0 0 -4 0 0

.

.

.

1 0 1 0 -1 , 0 1 2 0 3 , 0 0 0 0 1

A basis for the column space is

A basis for the nullspace is

.

1 0 1 0 -1 , 0 1 2 0 3

A basis for the column space is

A basis for the nullspace is

1 0 0 , 1 0 0

1 0 0 0 , 1 , 0 0 0 1

-1 1 -2 -3 1 , 0 0 -4 0 0

.

.

.

16

.

70)

D) A basis for the row space is

1 0 1 0 -1 , 0 1 2 0 3 , 0 0 0 1 4

A basis for the column space is

A basis for the nullspace is

1 0 0 0 , 1 , 0 0 0 1

-1 1 -2 -3 1 , 0 0 -4 0 0

.

.

.

71) Find the dimensions of the nullspace and the column space of the given matrix.

A=

1 -2

-5 3

-4 3 -1 -4

0 1

A) dim Null A = 4, dim Col A = 1 C) dim Null A = 3, dim Col A = 2

B) dim Null A = 2, dim Col A = 3 D) dim Null A = 3, dim Col A = 3

72) Find the dimensions of the nullspace and the column space of the given matrix.

1 -2 A= 0 0 0 0 0 0

3 1 1 -6 0 0 0 0

0 2 0 0

71)

5 -4 -2 0 1 3 0 0

A) dim Nul A = 3, dim Col A = 4 C) dim Nul A = 5, dim Col A = 2

B) dim Nul A = 4, dim Col A = 3 D) dim Nul A = 2, dim Col A = 5

17

72)

73) Find a basis for Col B where

73)

1 -1 0 0 B= 0 0 0 0 0 0

A)

1 0 0 0 0 0 0 , 1 , 0 0 0 1 0 0 0

C)

1 0 0 0 0 1 0 0 0 , 0 , 1 , 0 0 0 0 1 0 0 0 0

3 -2 -4 -6

-1 74) Let A = 1 2 3

7 -7 -9 -11

0 -2 1 4 0 0 0 0 0 0

2 -1 -5 -9

0 0 1 0 0

0 0 0 .

1 0

0 1 -3 3 and B = 0 1 1 0 0 -1 0 0

B)

1 0 0 , 0 0

-1 0 0 1 0 , 0 , 0 0 0 0

D)

1 0 0 0 0 , 1 , 0 , 0 0 0 1 0 0 0 0 1

-7 0 5 0

-2 1 -3 0

-2 0 0 4 0 0 0 , 1 , 0 0 0 1 0 0 0

0 3 .

-5 0

74)

It can be shown that matrix A is row equivalent to matrix B. Find a basis for Col A.

A)

3 7 -1 1 , -2 , -7 2 -4 -9 3 -6 -11

C)

1 -3 -7 0 , 1 , 0 0 0 5 0 0 0

B)

3 2 -1 1 , -2 , -1 2 -4 -5 3 -6 -9

D)

3 7 2 0 -1 1 , -2 , -7 , -1 , 3 2 -4 1 -9 -5 3 -6 -11 -9 -1

75) Find the inverse of the matrix. A = -1 -6 6

A)

1 11

-

2 11

2 11

-

1 33

1 11

B)

-

75)

3 2 11

-

C)

2 1 11 33

18

2 1 11 33 1 11

2 11

D)

-

1 33

2 11

-

2 11

1 11

76) Find the inverse of the matrix. A = 0 5

76)

6 3

1 6

0

A) 1

-

5

B)

1 10

-

1 5

0

1 10

1 6

-

C)

1 10

-

-

1 6

1 5

0

-

D)

1 10

1 6

1 5

0

77) Find the inverse of the matrix, if it exists. A = -4 0

77)

-3 -3 -

A) A is not invertible.

-

C)

1 4 1 4

B)

0 -

D)

1 3

78) Find the inverse of the matrix, if it exists. A =

1 3 1 4

-

1 4

-

1 4

-

1 4 0

-

1 3

78) 2 9

-

B) 1

-

6

C)

-

3 4 -6 -8

A) A is not invertible.

2 9

0

1 9

-

D)

1 1 6 12

1 6

1 9

1 12

-

2 9

1 12 1 9

79) Find the inverse of the matrix. A = -3 1

79)

0 6

-

A)

1 3 0

-

1 18 1 6

-

B)

1 3

1 18

0

1 6

1 6

C)

19

1 18

0 -

1 3

D)

-

0

1 6

1 3

1 18

80) Find the inverse of the matrix. A = -2 -5

80)

1 0

A)

0 1 1 2 5 5

B)

1 2 5 5 0

C)

1

-

2 1 5

-

1 0 5

D)

0 1 5

-1 2 5

1 0 0

81) Find the inverse of the matrix. -1 1 0

81)

1 1 1

1 -1

1

0

1

1 1 1

0

-1

B) 0 1 1

A) 0 1 -1

0

0

C) -1 -1 0

0 0 1

D)

-1 -1 -1

5 -1 5 0 4 10 -1 9

82) Find the inverse of the matrix, if it exists. A = 5

5 5 10

A) A-1 = -1 0 -1 5 4

C) A-1 =

1 0

82)

B) A-1 does not exist.

9

1

4 5

0

4 5

D) A-1 = 0 1 -1

0 1 -1 0 0 0

0

20

1 0 0 1 1 0 -2 -1 1

4 5

0

1 1 1

83) Find the inverse of the matrix, if it exists. A = 2 1 1

83)

2 2 3

-1 -1 -1

A) A-1 = -2 -1 -1

B) A-1 =

-2 -2 -3

C) A-1 =

1 1 2

1 1

1 2

1 1 2 3

1 1

-1 1 0 4 -1 -1 -2 0 1

D) A-1 does not exist.

1 3 2

84) Find the inverse of the matrix, if it exists. A = 1 3 3

84)

2 7 8

B) A-1 =

A) A-1 does not exist.

1

1 3

1 2

C) A-1 = 1

1 3

1 3

1 2

1 7

1 8

-3 10 -3 2 -4 1 -1 1 0

-1 -3 -2

D) A-1 = -1 -3 -3 -2 -7 -8

1 0 8

85) Find the inverse of the matrix, if it exists. A = 1 2 3

85)

2 5 3

9 -40 16

1 1 2

A) A-1 = -3 13 -5 -1

B) A-1 = 0 2 5

5 -2

8 3 3 -1

0 -8

D) A-1 = -1 -2 -3

C) A-1 does not exist.

-2 -5 -3

21

6 -4 1

86) Find the inverse of the matrix, if it exists. A = 11 -7 2

86)

5 -3 1

1 6

1 11

1

1

1

1

1 1 5 3

1

A) A-1 = 11 - 7 2

1 11

2 -4 11

5

8 7

3

6 1 5 3

1

B) A-1 = 11

6 11

5

1

1

D) A-1 = -4 -7 -3

C) A-1 does not exist.

2

0 2 2

87) Find the inverse of the matrix, if it exists. A = -2 0 4

87)

0 7 0

0

1 2

0

0

B) A-1 = - 7

0

1 7

2 7

1 1 7 7

1 2

0

0

1 -

1 2 2 7

1 1

A) A-1 = - 2 -

C) A-1 = 0 1 2

0

1 7

0 -

1 7

-1 1

1 2 2 7

D) A-1 does not exist.

88) Let A be an n × n matrix such that A2 = 0. Then what can we say about the matrix I + A? A) I + A is invertible, and (I + A)-1 = I - A. B) I + A is invertible, and (I + A)-1 = I + A. C) I + A is not invertible.

D) I + A is invertible, and (I + A)-1 = I.

22

88)

89) Solve the system by using the inverse of the coefficient matrix.

89)

5x1 + 3x2 = 3 2x1 + 5x2 = 24

A) (-3, 6)

B) (-3, -6)

C) No solution

D) (6, -3)

90) Solve the system by using the inverse of the coefficient matrix.

90)

6x1 + 4x2 = 4 3x1 = -6

A) No solution

B) (-2, -4)

C) (4, -2)

D) (-2, 4)

91) Solve the system by using the inverse of the coefficient matrix.

91)

7x1 - 2x2 = 2 28x1 - 8x2 = 3

A) 2 - 7 t,t 7

2

B) No solution

C) (2, 3)

D) (4, 4)

92) Solve the system by using the inverse of the coefficient matrix.

92)

2x1 + 6x2 = 2 2x1 - x2 = -5

A) (1, -2)

B) (2, -1)

C) (-1, 2)

D) (-2, 1)

93) Solve the system by using the inverse of the coefficient matrix.

93)

2x1 - 6x2 = -6 3x1 + 2x2 = 13

A) (-3, -2)

B) (2, 3)

C) (3, 2)

D) (-2, -3)

94) Solve the system by using the inverse of the coefficient matrix.

94)

10x1 - 4x2 = -6 6x1 - x2 = 2

A) (-1, -4)

B) (-4, -1)

C) (4, 1)

23

D) (1, 4)

95) Solve the system by using the inverse of the coefficient matrix.

95)

2x1 - 4x2 = -2 3x1 + 4x2 = -23

A) (-5, -2)

B) (2, 5)

C) (-2, 5)

D) (5, 2)

96) Solve the system by using the inverse of the coefficient matrix.

96)

-5x + 3y = 8 2x - 4y = -20

A) (-2, -6)

B) (-6, -2)

C) (2, 6)

D) (6, 2)

97) Solve the system by using the inverse of the coefficient matrix.

97)

x1 - x2 + 3x3 = -8 2x1 + x3 = 0 x1 + 5x2 + x3 = 40

A) (8, 8, 0)

B) (0, -8, -8)

C) (0, 8, 0)

D) (-8, 0, 0)

98) Determine which of the following linear mappings is invertible.

98)

Let f A be a linear mapping whose standard matrix is 1 -2 3 A = -1 3 -4 .

2 -2 -9

Let T be a linear mapping defined by T(x1 , x2 , x3 ) = (-2x2 - 2x3 , -2x1 + 8x2 + 4x3 , -x1 - 2x3 , 4x2 + 4x3 ).

A) fA only

B) T only

C) Both

D) None

99) Write a 4 × 4 elementary matrix that corresponds to swapping the third and fourth rows.

A)

1 0 0 0 0 1 0 0 0 0 0 1 0 0 -1 0

B)

1 00 0 0 10 0 0 00 1 0 01 0

C)

24

1 00 0 0 10 0 0 01 1 0 00 1

D)

1 00 0 0 10 0 0 01 0 0 01 0

99)

100) Write a 4 × 4 elementary matrix that corresponds to multiplying the second row by 1 .

2

A)

1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0

B)

1 0 0 0 1 0 0 0 2

C)

0 0 1 0 0 0 0 1

1 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 2

D)

100)

1 0 0 0 0 1 0 0 1 0 0 0 2 0 0 0 1

101) Determine if matrix A is an elementary matrix, and if so, determine the corresponding row

101)

operation.

A=

1 0

0 0 1 0

0 0 1 1 3

0

0

0

0

0

1

A) A is the elementary matrix obtained from I4 by adding 1 times the fourth row to the third.

3

B) A is the elementary matrix obtained from I4 by adding - 1 times the fourth row to the third.

3

C) A is the elementary matrix obtained from I4 by multiplying third row by - 1 .

3

D) A is not an elementary matrix.

102) Determine if matrix A is an elementary matrix, and if so, determine the corresponding row operation.

0 A= 0 1

1 0 0

0 1 0

A) A is the elementary matrix obtained by switching the first, second and third rows.

B) A is the elementary matrix obtained by switching the first and third rows.

C) A is not an elementary matrix.

D) A is the elementary matrix obtained by switching row one with row two.

25

102)

103) Determine if matrix A is an elementary matrix, and if so, determine the corresponding row

103)

operation.

0 A= 1 0

1 0 0

0 0 1

A) A is the elementary matrix obtained by switching the first, second and third rows.

B) A is the elementary matrix obtained by switching row one with row two.

C) A is not an elementary matrix.

D) A is the elementary matrix obtained by switching the first and third rows.

104) Express A = -2 1 as a product of elementary matrices.

1

104)

3

A) 1 3 1 0 1 0 0 1

B) 0 1 1 0 1 0 1 3

C) 0 1 1 0 1 0

D) 1 -3 1 0

0 1 0 7 -2 1 1 0

1 0 -2 1 0 7 0 1

1 -3 1 0 2 1 0 1/7 0 1

0

230

1

105) Write the matrix A = 1 3 0 as a product of elementary matrices.

001

010

1 0 0

110

1 00

001

0 0 1

001

0 0 1

010

100

1 -1 0

1 0 0

A) A = 1 0 0 -2 1 0 0 1 0 0 -3 0

B) A = 1 0 0 2 1 0 0 1 0 0 -3 0 001 001 0 0 1 0 0 1 1 0 0 1 -1 0 1 0 0 0 1 0 C) A = 0 -3 0 0 1 0 2 1 0 1 0 0 0 0 1 0 0 1 001 001 010 D) A = 1 0 0 001

1 0 0 -2 1 0 0 0 1

110 010 001

1

0 1 0 3 0

0

0 0 1

26

1 0 0 1 0 1/7 2 1 1 0

105)

106) Find a LU-decomposition for matrix A =

3 -1 .

-18 9

106)

A) A = 1 0

-3 -1 0 -3

B) A =

1 0 -6 1

3 -1 0 3

C) A = 1 0

-6 -1 0 3

D) A = 1 0

3 1 0 -3

6 1

3 1

-6 1

107) Find a LU-decomposition for matrix A = 3 -1

107)

9 -5

A) 1 0 3 -1

B) 1 0 3 -1

C) 3 -1 1 0

D) 3 -1 -1/2 1/2

-3 1 0 -2

3 1 0 -2

0 -8 3 1

0 -2 -9/2 5/2

2

4

5 5 .

4 -1 24

108) Find a LU-decomposition for matrix A = 4 11 1

0 0

A) A = 2 1 0 2 -3 1 1

0 0

C) A = 2 1 0 2 -3 1

108)

3 4 5 0 -3 5 0 0 1

B) A = 4

2 4 5 0 3 -5 0 0 -1

D) A = 4

1

0 0 1 0 4 -1 1

2 4 5 0 11 5 0 0 24

1

2 4 5 0 3 -5 0 0 -1

0 0 1 0 4 -1 1

109) Solve the equation Ax = b by using the LU-factorization given for A. Given A =

109)

-2 1 2 1 0 -2 1 ,b= , and A = .

2 5 3 -1 1 0 6

A) -7 , 5 12 6

C) -3 , 1

B) (2,5)

4 2

27

D) -3 , -1 4

4

110) Solve the equation Ax = b by using the LU-factorization given for A.

110)

3 -1 2 6 Given A = -6 4 -5 , b = -3 and 9 5 6 2 1 0 0 A = -2 1 0 3 4 1

3 -1 2 0 2 -1 .

0 0 4

49

A) x = -38

32

10

25

B) x = -2

C) x = -58

51

-13

22

D) x = -7

15

111) Solve the equation Ax = b by using the LU-factorization given for A. 1 2 4 3 -1 -3 -1 -4 , b = Given A = 2 1 19 3 1 5 -9 7 1 0 00 -1 1 00 A= 2 3 10 1 -3 -2 1

27 A) x = 9 8 -3

111)

2 0 , and 4 3

1 24 3 0 -1 3 -1 .

0 02 0 0 00 1

41 -6 B) x = -3 -5

27 -18 C) x = 89 -13

28

2 -2 D) x = 8 -3

Answer Key Testname: UNTITLED3

1) D 2) B 3) D 4) D 5) B 6) B 7) C 8) D 9) B 10) C 11) D 12) C 13) C 14) B 15) C 16) D 17) A 18) C 19) D 20) B 21) D 22) C 23) C 24) A 25) C 26) C 27) D 28) B 29) B 30) C 31) D 32) D 33) C 34) B 35) C 36) C 37) D 38) B 39) C 40) B 41) C 42) D

29

Answer Key Testname: UNTITLED3

43) B 44) B 45) A 46) A 47) D 48) B 49) B 50) A 51) B 52) B 53) D 54) A 55) B 56) C 57) A 58) C 59) B 60) A 61) C 62) C 63) D 64) A 65) D 66) A 67) B 68) A 69) B 70) D 71) C 72) B 73) C 74) A 75) B 76) D 77) C 78) A 79) B 80) A 81) D 82) B 83) B 84) B

30

Answer Key Testname: UNTITLED3

85) A 86) C 87) C 88) A 89) A 90) D 91) B 92) D 93) C 94) D 95) A 96) C 97) C 98) A 99) B 100) B 101) B 102) C 103) B 104) B 105) B 106) B 107) B 108) C 109) A 110) B 111) B

31

Exam Name___________________________________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

1) Calculate 2(-1 + 2x - 3x 2) - 3(5 - 4x - x 3 ).

A) -17 + 16x - 6x2 + 3x3 C) 13 - 8x - 6x2 + 3x3

1) B) -17 - 8x - 6x2 + 3x3 D) 13 + 16x - 6x2 + 3x3

2) Calculate (-1 + 2x - 3x 2 - x 3) - 2(1 - x3 ).

A) -3 + 2x - 3x2 + x3 C) -3 + 2x - 3x2 - 3x3

2) B) 1 + 2x - 3x2 + x3 D) 3 + 2x - 3x2 + 2x3

3) Determine whether polynomials p(x) and q(x) are in the span of = {1 + x,x + x 2 , 1 - x 2 } where

3)

p(x) = 1 - 2x + x 2 , q(x) = 3 + x 2 .

A) q(x) only C) Both p(x) and q(x)

B) p(x) only D) Neither p(x) nor q(x)

4) Determine whether polynomials p(x) and q(x) are in the span of = {1 + x + x 2, 1 + x - x 2} where

4)

p(x) = 1 - x + x 2 , and q(x) = 3 + 3x + x 2 .

A) p(x) only C) Both p(x) and q(x)

B) q(x) only D) Neither p(x) nor q(x)

5) Determine whether polynomials p(x) and q(x) are in the span of = {1 + x, x + x 2, 1 - x 3 } where p(x) = 2, and q(x) = 3 + x 3 .

A) q(x) only

5)

B) p(x) only D) Neither p(x) nor q(x)

C) Both p(x) and q(x)

6) Determine whether polynomials p(x) and q(x) are in the span of = {1 + x, x + x 2, 1 - x 3 } where p(x) = 3 - x2 - 2x3 , and q(x) = 3 + x3 .

A) q(x) only C) Both p(x) and q(x)

B) p(x) only D) Neither p(x) nor q(x)

1

6)

7) Determine whether the following sets of vectors {p 1 , p 2 , p3 } are linearly independent.

7)

A: p 1 (t) = 1, p 2 (t) = t2 , p 3 (t) = 2 + 3t B: p 1 (t) = t, p 2 (t) = t2 , p 3 (t) = 2t + 3t2 C: p 1 (t) = 1, p 2 (t) = t2 , p 3 (t) = 2 + 3t + t2

A) A only B) A and C C) C only D) B only E) All of them 8) Determine whether the sets A and B are linearly independent sets of polynomials

8)

A = 1 + 2t, 3 + 6t2 , 1 + 3t + 4t2 , B = 1 + 2t + t2 , 3 - 6t2 , 1 + 4t + 4t2

A) A only C) Both A and B

B) B only D) Neither A nor B

9) Determine whether the sets A and B are linearly independent sets of polynomials

9)

A = {1 + 2x + x2 - x3 , 5x + x2 , 1 - 3x + 2x2 + x3 } , B = {1 + x + x2 , x, x2 + x3 , 3 + 2x + 2x2 - x3 }

A) A only C) Both A and B

B) B only D) Neither A nor B

10) The following set: 1 + 2t2, 4 + t + 5t2 , 3 + 2t is linearly dependent. Find a linear combination that equals the zero vector.

A) -5(1 + 2t2 ) + 2(4 + t + 5t2 ) - (3 + 2t) = 0

10)

B) -9(1 + 2t2 ) + 3(4 + t + 5t2 ) - (3 + 2t) = 0 D) -5(1 + 2t2 ) - 2(4 + t + 5t2 ) - (3 + 2t) = 0

C) -3(1 + 2t2 ) - 4(4 + t + 5t2 ) + (3 + 2t) = 0

11) Determine which of the following sets is a subspace of Pn for an appropriate value of n.

A: All polynomials of the form p(t) = a + bt2 , where a and b are in B: All polynomials of degree exactly 4, with real coefficients C: All polynomials of degree at most 4, with positive coefficients A) A only B) B only C) C only

2

D) Both A and B

11)

12) Let H be the set of all polynomials having degree at most 4 and rational coefficients. Determine

12)

whether H is a subspace of 4 . If it is not a subspace, determine which of the following properties it fails to satisfy.

A: Contains zero vector B: Closed under vector addition C: Closed under multiplication by scalars A) H is not a subspace; does not contain zero vector.

B) H is not a subspace; not closed under vector addition.

C) H is a subspace.

D) H is not a subspace; not closed under multiplication by scalars.

13) Let H be the set of all polynomials of the form p(t) = a + bt2 where a and b are in and b > a.

13)

Determine whether H is a subspace of 2 (R). If it is not a subspace, determine which of the following properties it fails to satisfy.

A: Contains zero vector B: Closed under vector addition C: Closed under multiplication by scalars A) H is not subspace; not closed under vector addition.

B) H is not a subspace; not closed under multiplication by scalars and does not contain zero vector.

C) H is not a subspace; not closed under multiplication by scalars.

D) H is not a subspace; does not contain zero vector.

14) Let H be the set of all points of the form (s, s -1). Determine whether H is a subspace of R2 . If it is

14)

not a subspace, determine which of the following properties it fails to satisfy.

A: Contains zero vector B: Closed under vector addition C: Closed under multiplication by scalars A) H is not a subspace; does not contain zero vector.

B) H is a subspace.

C) H is not a subspace; fails to satisfy all three properties.

D) H is not a subspace; not closed under vector addition. x y x + 2y = 0, x,y,z,w . Determine whether W is a subspace of M2×2 (R). If it is zw not a subspace, determine which of the following properties it fails to satisfy.

A: Contains zero vector B: Closed under vector addition C: Closed under multiplication by scalars A) H is not a vector space; not closed under vector addition.

15) Let W =

B) H is not a vector space; does not contain zero vector.

C) H is not a vector space; fails to satisfy all three properties.

D) H is a vector space.

3

15)

x x + 2 x,w . Determine whether W is a subspace of M2×2 (R). If it is not a 0 w subspace, determine which of the following properties it fails to satisfy.

A: Contains zero vector B: Closed under vector addition C: Closed under multiplication by scalars A) H is not a subspace; does not contain zero vector.

16) Let W =

16)

B) H is not a subspace; not closed under vector addition and does not contain zero vector.

C) H is not a subspace; fails to satisfy all three properties.

D) H is a subspace.

17) Determine which of the following sets are a basis for M2×2 (R).

B=

1 0 , 0 1 , 0 0 , 0 0 0 0 0 0 1 0 0 1

,C=

17)

1 0 , 0 1 , 1 1 , 0 1 0 1 1 0 1 0 0 -1

A) Neither B nor C C) Both B and C

B) C only D) B only

18) Determine which of the following sets are a basis for R3 .

1 2 -3 B = v1 = -3 , v2 = 8 , v3 = -2 -6 -4 4

, C = v1 =

18)

1 4 -2 4 , v2 = 0 , v3 = -4 -5 -14 4

A) Neither B nor C C) Both B and C

B) C only D) B only

19) Determine which of the following sets are a basis for 3(R).

19)

B = 1 + x, 1 + x + x2 , 1 + x3 , C = 1 + x, 1 - x, 1 + x3, 1 - x3

A) Neither B nor C C) Both B and C

20) For the subspace 1

-2

3

0

s - 2t s+t 3s ; s,t

B) C only D) B only

R , find a basis and state the dimension.

1

-2

3

3

1

-2

3

0

A) 1 , 1 dim2

B) 1 , 1 dim2

-1 2 , dim1 3

D) 1 , 1 dim3

C)

4

20)

21) Given the set of vectors

1 0 0 , 1 0 2

, decide which of the following statements is true:

21)

A: Set is linearly independent and spans 3 . Set is a basis for 3 .

B: Set is linearly independent but does not span 3 . Set is not a basis for 3 .

C: Set spans 3 but is not linearly independent. Set is not a basis for 3 .

D: Set is not linearly independent and does not span 3 . Set is not a basis for 3 .

A) A B) B C) C D) D

22) Given the set of vectors

1 0 0 0 0 , 1 , 0 , 1 0 0 1 1

, decide which of the following statements is true:

22)

A: Set is linearly independent and spans 3 . Set is a basis for 3 .

B: Set is linearly independent but does not span 3 . Set is not a basis for 3 .

C: Set spans 3 but is not linearly independent. Set is not a basis for 3 .

D: Set is not linearly independent and does not span 3 . Set is not a basis for 3 .

A) A

B) B

a + 3b + 4d c+d 23) Let H = : a, b, c, d in -3a - 9b + 4c - 8d -c - d Find the dimension of the subspace H. A) dim H = 4 B) dim H = 2

C) C

D) D

.

23)

C) dim H = 1

D) dim H = 3

24) Determine which of the following statements is false.

24)

A: The dimension of the vector space p 7 of polynomials is 8.

B: Any line in R3 is a one-dimensional subspace of R3 .

C: If a vector space V has a basis B = b1 , ......, b3 , then any set in V containing 4 vectors must be linearly dependent.

A) A

B) B

C) C

D) A and B

25) Determine which of the following statements is true.

25)

A: If V is a 4-dimensional vector space, then any set of exactly 4 elements in V is automatically a basis for V. B: If there exists a set v 1 , ......, v 5 that spans V, then dim V = 5.

C: If H is a subspace of a finite-dimensional vector space V, then dim H dim V. A) A B) B C) C D) A and C

a b : a, b, c in . bc Find the dimension of the subspace H. A) dim H = 1 B) dim H = 2

26) Let H =

26) C) dim H = 4

5

D) dim H = 3

27) Let H = a + bx + cx2

2 : a = -c .

Find the dimension of the subspace H. A) dim H = 4 B) dim H = 3

27) C) dim H = 2

D) dim H = 1

28) Let C= 1 0 , 1 1 . Extend C to a basis for M2×2 .

01

28)

00

A)

10 , 11 , 01 , 00 01 00 00 10

B)

10 , 11 , 01 01 00 00

C)

10 , 11 , 00 01 00 10

D)

10 , 11 , 10 , 01 01 00 00 00

29) Extend a basis

-7 4 1 , 0 0 1

for the plane -2x - 14y + 8z = 0 to obtain a basis for 3 .

A)

-7 4 0 1 , 0 , 1 0 1 0

B)

-7 0 1 , 1 0 0

C)

4 0 0 , 1 1 0

D)

-7 4 -3 1 , 0 , 1 0 1 1

29)

30) Determine a basis B for the plane x -2y + 3z = 0.

2

A) 1 , 0

30)

1

-3 0 1

1

0

0

0

0

3

C) 0 , -2 , 0

B) -2 3

2

6

0

1

D) 1 , 3

31) Find the vector x determined by the given coordinate vector [x]B and the given basis B. B=

A)

1 , 0 3 1

4 -8

, [x]B = -4 4

B) 1

C) -4

4

-8

6

D) -4 4

31)

32) Find the coordinate vector [x]B of the vector x relative to the given basis B. b1 = 1 , b2 = 1 , x = 1 , and B = -4 2 8

A)

9 -30

32)

b1 , b2

B) 2

C)

0

2 -1

D) 1 8

33) Find the coordinate vector [x]B of the vector x relative to the given basis B. b1 =

3 4 6 5 , b2 = -3 , x = -19 , and B = -4 -2 2

42

A) -38

b1 , b2

C) -2

B) -2 3

-12

33)

2

D)

-58 87 14

34) Find the coordinate vector [x]B of the vector x relative to the given basis B.

34)

B = 1 + x + x 2 , 1 + 3x + 2x 2 , 4 + x2 and x = -2 + 4x + 2x 2

A)

1 1 -1

1

B) -1

C)

1

-1 1 1

-1

D) -1 1

35) Find the coordinate vector [x]B of the vector x relative to the given basis B. B=

A)

10 , 01 , 11 , 0 1 0 1 1 0 1 0 0 -1

, x= 0 1 12

-2 3 2 0

2 3 -2 0

B)

C)

-2 3 -2 0

35)

D)

2 -3 2 0

36) Consider two bases B = b1 , b2 and C = c1 , c2 for a vector space V such that b1 = c1 - 6c2 and b2 = 4c1 + 3c2 . Suppose x = b1 + 2b2 . That is, suppose [x]B = 1 . Find [x]C .

2 A) 8 B) 9 C) -11 D) 6 0 0 10 -9

7

36)

37) Consider two bases B = b1 , b2 , b3 and C = c1 , c2, c3 for a vector space V such that

37)

b1 = c1 + 2c3 , b2 = c1 + 4c2 - c3 , and b3 = 3c1 - c2 . Suppose x = b1 + 6b2 + b3 . That is, 1 suppose [x]B = 6 . Find [x]C. 1

10

10

3

A) 25

B) 24

-6

-3

C) 23 8

10

D) 23 -4

38) Let B = b1 , b2 and C = c1 , c2 be bases for R2 , where b1 =

38)

1 , b = -3 , c = 1 , c = -4 .

2 1 2 -3 -10 3 3

Find the change-of-coordinates matrix from B to C.

A)

1 -3 -3 6

B)

-11 -3

-2

21 6

C)

7 11 -1 3

D) 1 -4

3 -10

39) Consider two bases B = b1 , b2 and C = c1 , c2 for a vector space V such that

39)

b1 = c1 - 2c2 and b2 = 3c1 - 4c2 . Find the change-of-coordinates matrix from B to C.

B) 1 3

A) 1 -2 3 -4

C)

24

1 3 -2 -4

D)

0 3 -2 -4

40) Determine which of the following mappings are linear. : 2

P1

defined by

: M(2,2)

defined by

A) Neither nor T

40)

a = (a + b) + bx. b a b = a + d. cd B) only

C) only

D) Both and

41) Determine which of the following mappings are linear. : 2

M(2,2)

: M(2,2)

P3

defined by

defined by

A) Neither nor

41)

a = a1 . b 1b a b = cx + (a + b) x3 cd B) only C) only

8

D) Both and

42) Let : 2 M(2, 2) be a linear mapping defined by

a + bx + cx2 =

Determine whether the following matrices are in the range of .

a+b 0

0 . b+c

42)

A = 1 0 and B = 3 0 0 1 0 2

A) Both A and B C) A only

B) Neither A nor B D) B only

a b = a-b b-c . c d c-d d-a Determine whether the following matrices are in the range of .

43) Let : M(2, 2) M(2, 2) be linear mapping defined by

43)

A = 1 0 and B = 0 0 0 1 0 1

A) Both A and B C) A only

B) Neither A nor B D) B only

44) Find a basis for the null space of the linear mapping : M(2, 2) M(2, 2) defined by

44)

a b = a-b b-c . cd c-d d-a

A)

1 1 1 0

B)

1 -1 -1 1

C)

1 1 1 1

D)

0 1 1 1

45) Find a basis for the range of the linear mapping : M(2, 2) M(2, 2) defined by

45)

a b = a-b b-c . c d c-d d-a

A)

1 1 , 1 -1 , 0 -1 1 0 -1 1 1 0

B)

0 1 , -1 1 1 1 0 0

C)

1 0 , -1 1 , 0 -1 , 0 -1 0 0 1 0

D)

1 -1 -1 1

46) Find a basis for the null space of the linear mapping : M(2, 2)

defined by

A)

1 0 , 0 1 , 0 0 0 -1 0 0 1 0

B)

1 0 , 0 1 0 -1 0 0

C)

1 0 0 -1

D)

1 0 , 0 0 0 -1 1 0

9

a b = a + d. c d

46)

47) Find a basis for the range of the linear mapping : M(2, 2) A) {0}

B) {1,x}

defined by

C) no basis

a b c d

= a + d.

D) {1}

48) Determine the matrix of a linear mapping with respect to a given basis B. Let L: 3 B=

47)

48)

3 be a linear mapping defined by L( v 1 ) = v 2 , L( v 2 ) = v 1 , L( v 3 ) = 5 v 1 and

v 1 , v 2 , v 3 . Find the B-matrix of the linear operator L: V

0 1 0

0 0 0

A) 1 0 0

0 0 5

0 1 5

C) 1 0 0

B) 1 1 0

0 0 5

V relative to B.

0 0 5

D) 1 1 0 0 0 0

0 0 0

49) Let : 3

3 be a linear mapping defined by (1,0,1) = (1,2,1), (1,-1,0) = (0,1,-1), and (0,1,-1) = (3,-3, 0), where 1 1 0 B = 0 , -1 , 1 .

1 0 -1

2 0 0

2 0 3

A) -1 1 3

2 0 0

B) -1 0 0

1 0 0

C) -1 0 3

1 1 0

2 1 . Given a natural basis B = -1 transformation

50) Let v =

D)

1 1 0

2 1 0 1 , 0 , 1 -1 2 1

, determine B-matrix of the

proj v .

A) [proj v ] =

1 0 0

B) [proj v ] =

0 1 0

0 1 0 0 0 0

0 0 0

C) [proj v ] =

0 0 0

1 0 0

D) [proj v ] =

0 0 0 0 0 0

-1 0 0 0 00 0 00

10

49)

2 0 0 1 0 3 -1 1 0

50)

51) Let : 3

3 1 1 3 be a linear mapping with the standard matrix 0 4 -2 .

1 -1 5

Determine matrix of with respect to the basis B = 042

A) 4 2 0 004

B)

400

1 0 1 1 , 1 , 0 0 1 1

C)

042 420

51)

.

004

420

420

D) 0 4 2

042

004

2 be a linear mapping with the standard matrix A = -67 -60 . Determine matrix of 72 65 1 , 5 . with respect to the basis B = -1 -6 A) 7 0 B) 7 0 C) -7 0 D) -7 0 0 -5 0 5 0 5 0 -5

52) Let : 2

2 be a linear mapping with the standard matrix A = -232 -1156 . Determine matrix 49 244 of with respect to the basis B = 34 , -5 .

-7 1 6 1 1 -6 A) B) C) 6 -1 D) 6 1 0 7 0 -6 0 6 0 6

53) Let : 2

54) Let : 3

3 be a linear mapping with the B-matrix

Determine (0.1,0).

A) (3, 1, -2)

55) Let : 2

0 4 2

, where basis B =

0 0 4

B) (1, 4, 1)

C) (4, 1, -1)

2 be a linear mapping with the B-matrix

Determine (0,1).

A) (3, -5)

4 2 0

B) (-2, 4)

13 2 21 2

1 2

1 , where basis B = 2

C) (3, 5)

11

1 0 1 1 , 1 , 0 0 1 1

.

52)

53)

54)

D) (-1, 4, -1)

3 , -1 1 1

D) (2, 4)

.

55)

56) Suppose B = {b1 , b2 } is a basis for V, and C = {c1 , c2, c3 } is a basis for W.

56)

Let T be defined by T(b1 ) = -5c1 - 6c2 + 5c3 T(b2 ) = -5c1 - 12c2 + 2c3 Find the matrix of the linear mapping T: V

A) -5 -6 5 0

6 3

B)

W relative to B and C. -5 0 -5 -6 5 C) -6 -6 -5 -12 2 5 -3

-5

-5

5

2

D) -6 -12

57) Suppose B = {b1 , b2 } is a basis for V, and C = {c1 , c2, c3 } is a basis for W.

57)

Let T be defined by T(b1 ) = -5c1 - 6c2 + 5c3 T(b2 ) = -5c1 - 12c2 + 2c3 Find the matrix of the linear mapping T: V -5 -5 -5 0 A) -6 -12 B) -6 -6 5 2 5 -3

W relative to B and C.

C) -5 -6 5 0

6 3

-6 5 -5 -12 2

D) -5

58) Determine which of the following pairs of vector spaces are isomorphic.

I) P = p(x)

2 : p(2) = 0 and T =

II) P2 and R2

A) Neither I nor II

a 0 0 b

M2×2 : a,b

B) II only

58)

.

C) I only

D) Both I and II

59) Which of the following statements are true?

59)

I) If the vector spaces U and V have the same finite dimension then they are isomorphic.

II) Any plane through the origin in 3 is isomorphic to 2 .

A) I only

B) Both I and II

C) II only

D) Neither I nor II

60) Let T be the linear transformation whose standard matrix is 1 -2 3 A = -1 3 -4 .

2 -2 -9

Determine whether the linear transformation T is one-to-one and whether it maps 3 onto 3 .

A) One-to-one; onto 3 B) Not one-to-one; not onto 3

C) Not one-to-one; onto 3

D) One-to-one; not onto 3

12

60)

61) T(x1 , x2 , x3 ) = (-2x2 - 2x 3 , -2x1 + 8x 2 + 4x3 , -x 1 - 2x 3, 4x2 + 4x 3 ) Determine whether the linear transformation T is one-to-one and whether it maps 3 onto 4 .

A) Not one-to-one; not onto 4 B) One-to-one; onto 4

C) Not one-to-one; onto 4

D) One-to-one; not onto 4

13

61)

Answer Key Testname: UNTITLED4

1) A 2) A 3) D 4) B 5) D 6) B 7) B 8) A 9) A 10) A 11) A 12) D 13) B 14) C 15) D 16) C 17) C 18) D 19) A 20) A 21) B 22) C 23) B 24) B 25) C 26) D 27) C 28) A 29) A 30) A 31) C 32) C 33) B 34) A 35) B 36) B 37) D 38) B 39) C 40) D 41) B 42) A

14

Answer Key Testname: UNTITLED4

43) B 44) C 45) C 46) A 47) D 48) C 49) C 50) C 51) D 52) C 53) D 54) A 55) C 56) D 57) A 58) C 59) B 60) A 61) A

15

Exam Name___________________________________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

2 1 1

1) Let A = 1 2 1 . Evaluate the determinant of A by expanding along the first row.

1)

1 1 2

A) 2

B) 3

C) 4

D) 1

1 2 3

2) Let A = 4 5 6 . Evaluate the determinant of A by expanding along the first row.

2)

7 8 9

A) 3

B) 1

5 0 3) Find the determinant of the matrix, A = 0 0 0 A) -100 B) 0

C) -24 2 -2 5 -4 4 2 1 0 0 -2 3 7 .

0 0 -1 -5 0 0 0 -5 C) 200

D) 0

3) D) -200

4) Evaluate the determinant by using a cofactor expansion along any row or column.

-5 5 -5 3 0 -1 2 -2 0 3 0 0 0 -3 1 4 A) -90

B) -150

C) -30

D) 150

5) Evaluate the determinant by using a cofactor expansion along any row or column.

8 0 0 7 -4 0 -4 2 2 A) -64

B) 50

C) -78

B) -57

C) -48

1

5)

D) 64

6) Evaluate the determinant by using a cofactor expansion along any row or column.

6 -9 7 0 1 1 0 0 -8 A) 39

4)

D) 48

6)

7) Evaluate the determinant by using a cofactor expansion along any row or column.

-1 2 -1 4 3 -2 -3 -4 -1 A) 24

B) 14

C) -38

D) 38

8) Evaluate the determinant by using a cofactor expansion along any row or column.

3 5 2 1 2 2 3 4 3 A) 5

C) 55

B) -55

B) 39

C) -39

1 0 0 0 10) Determine the determinant of the elementary matrix 0 0 1 0 .

0 1 0 0 0 0 0 1 A) 2 B) -1 C) 0

10) D) 1

1 0 0

A) -k

B) 0

11) D) k

1 0 0

12) Determine the determinant of the elementary matrix 0 1 0 .

A) 4

0 0 -4 C) -4

B) 1

1 0 0 0 13) Determine the determinant of the elementary matrix 0 1 0 0 .

0 0 1 k 0 0 0 1 A) -1 B) 1 C) -k 1 0 0 5 14) Determine the determinant of the elementary matrix 0 1 0 0 .

0 0 1 0 0 0 0 1 A) -1 B) -5 C) 1

2

9)

D) -459

11) Determine the determinant of the elementary matrix 0 k 0 .

0 0 1 C) 1

8)

D) -5

9) Evaluate the determinant by using a cofactor expansion along any row or column.

7 5 9 4 5 7 6 3 4 A) 955

7)

12) D) 0

13) D) k

14) D) 5

1 k 0 0 15) Determine the determinant of the elementary matrix 0 1 0 0 .

0 0 1 0 0 0 0 1 A) -1 B) 1 C) -k

15) D) k

1 0 0 0 16) Determine the determinant of the elementary matrix 0 1 0 0 .

0 k 1 0 0 0 0 1 A) 1 B) -1 C) -k

16) D) k

0 1 0 0 17) Determine the determinant of the elementary matrix 1 0 0 0 .

0 0 1 0 0 0 0 1 A) 2 B) -1 C) 0

17) D) 1

1 0 0 0 18) Determine the determinant of the elementary matrix 0 1 0 0 .

0 0 1 0 0 0 0 k A) 1 B) k C) -k

18) D) -1 4

1

2

8

5

3

19) Find the determinant by row reducing to upper triangular form: -4 -1 -5 A) 36

B) 0

C) -12

19) D) -36

0 2

0

3 0

2

20) Find the determinant by row reducing to upper triangular form: 0 0 -5 A) 30

C) 0

B) -30

21) Find the determinant by row reducing to upper triangular form: A) -28

B) 14

D) 15 1 2 5 4 8 12 -2 3 -1

C) 56

2 -2 22) Find the determinant by row reducing to upper triangular form: 2 2 6 -6 -2 2 A) -80 B) -40 C) 80

3

20)

21) D) 28

6 4 7 1 14 14 -6 1

D) -160

22)

1 2 -6 -6 23) Find the determinant by row reducing to upper triangular form: 1 1 4 6 A) 5 B) 10 C) -5

-1 3 -2 5 3 -1 -2 5

23)

D) -10

24) Use a combination of row operations and cofactor expansions to find the determinant of matrix: 2 -2 6 4 2 2 7 1 6 -6 14 14 -2 2 -6 1 A) -80

C) 80

B) -160

D) -40

25) Use a combination of row operations and cofactor expansions to find the determinant of matrix: 2 0 -4 -2

1 -3 6 -1 4 5 5 1 A) 80

24)

25)

1 6 4 3

B) 1

C) -1

D) 0

1 a a2 26) Let A = 1 b b2 . Then det(A) is ________.

26)

1 c c2

A) 0 C) a + b + c

B) (a - b)(b -c )(a - c) D) (a - b)(b - c)(c - a) 2

4

6

27) Determine if the matrix 8 10 12 is invertible. If so, find its inverse.

27)

14 16 18

-2

-4

-6

14

16

18

A) It is invertible and the inverse is -8 -10 -12 .

B) It is invertible and the inverse is

-2 -4 8 10 -14 -16

-6 12 .

-8

C) It is invertible and the inverse is

-2 8 14

-6 12 .

18

-4 10 16

D) It is not invertible.

1 p 0

28) For what values of p is the matrix 0 1 2 not invertible?

A) 5 2

B) - 5 2

-1 2 1

28)

C) - 3 2

4

D) 3 2

1

0 0 is invertible. If so, find the inverse.

0 3 1 0 0 1 0 0 2 A) It is invertible and the inverse is .

1 0 0 3

29) Determine if the matrix 0

0 2 0

B) It is invertible and the inverse is

C) It is invertible and the inverse is

1 0 0

0 2 0

-1

0 1 2

0 0

29)

0 0 .

3 0 0

0

-

1 3

.

D) It is not invertible.

1 2 -1 p -1 -1 -2 3 not invertible?

30) For what values of p is the matrix 1 2 3 -1 2 1 -2 5

A) 32

B) 16

C) 8

30) D) 47 5

31) If A is an n × n invertible matrix, then find det(ATA-1).

A) -1 B) 0 C) 1 D) not enough information to determine the determinant a b c

d-3a

32) Suppose that det d e f = 2, and A = e-3a g h i

A) 1

f-3a

g+d h+e i+f

B) 3

a b . Find det(2A-1 ). c C) 4

33) Let A and B be n × n invertible matrices, then det(B -1 AB) = ________.

A) 0 B) det(A) C) -det(A)

31)

32) D) 12 33) D) det(B)

34) Let A and B be 3 × 3 matrices with det(A) = 2 and det(B) = -3. Then find det(-A2 B).

A) 12 B) 6 C) -12 D) -6

5

34)

35) Suppose that A is a 4 × 4 matrix and det(A) = -3. If C is obtained from A by switching the second column and the first column, what is det(C)?

A) 0 B) 3

35)

D) 12

C) -3

36) Suppose that A is a 4 × 4 matrix and det(A) = -3. If B is obtained from A by multiplying the second

36)

1 column of A by , what is det(B)?

2

A) - 3

C) 3

B) 6

2

D) 3

2

37) Suppose that A is a 4 × 4 matrix and det(A) = -1. What is det(3AAT)?

A) 9 B) 3 C) 81

37) D) -3

38) Let A and B be 3 × 3 matrix with det(A) = 2 and det(B) = -3. Then find det(-A2 B).

A) 12 B) 6 C) -12 D) -6 3

1 0

5

0 1

39) Suppose that A-1 = -2 -4 3 . Then find C of A. 3 4

-1 2

5 4

-1

0

3 4

1 4

1

A) 4 0

3 -2

1

0

1

39) 3 5

-2 5

1

1

-4 5

0

3 5

1 5

B) 5 0

C) 1 -4 0 3

3 5

1 5

0

-2

-4 5

3 5

1

0

1 5

D) 5

40) Let A = 1 1 , find the inverse of A by finding adj(A).

40)

2 1

2 1 -1

A) -1

38)

B) -1 1

C) 1 1

2 -1

2 -1

6

D)

1 -1 -2 1

41) Let A= 1 k , find the inverse of A by finding adj(A).

41)

0 1

A) 1 -k 0

B) 1 0

1

1

C)

k 1

1 0 -k 1

D) 1 k 0 1

t 1

42) Let A = -1 1 0 , find the inverse of A by finding adj(A).

42)

1 1 0

0

1 2

1 2

0

-

1 2

1 2

A) 0

1 2

1 2

B) 0

-

1 2

1 2

1

1 t 2 2

1 t 2 2

1

1 t 2 2

0

-

1 2

1 2

0

-

C) 0

1 2

1 2

D) 0

1 2

1

1 t 2 2

1 t 2 2

1

1 t 2 2

-

-

-

-

1 2

1 t 2 2

-

1 2 1 2

-

1 t 2 2

1 3 1

43) Let A = 3 1 1 . Determine element (A-1 )13 by using the cofactor method.

A) 2

1 1 3

5

B) - 1

C) - 2

10

5

44) Assume A is an invertible 3 × 3 matrix. Compute det(-A- 2 (C of A)).

A) 0 B) -1 C) 1

43) D) 1

10

44) D) 2

45) Solve the system of equations using Cramer's Rule.

45)

2x1 + 6x2 = 32 4x1 + x2 = -2

A) (6, -2)

B) (-2, 6)

C) (-6, -2)

D) (-2, -6)

46) Solve the system of equations using Cramer's Rule.

46)

4x1 + 2x2 = 34

-2x1 + 3x2 = -5

A) (3, 7)

B) (7, 3)

C) (-7, -3)

7

D) (-3, 7)

47) Solve the system of equations using Cramer's Rule.

47)

2x1 + 3x2 = 28 3x1 - 2x2 = 3

A) (5, 6)

B) (-6, 5)

C) (-5, 6)

D) (6, 5)

48) Solve the system of equations using Cramer's Rule.

48)

2sx1 + 4x2 = -3 2x1 + sx2 = 4

A) s ± 2; x1 =

4s+ 3 -3s - 16 and x2 = 2(s - 2)(s + 2) 2(s - 2)(s + 2)

B) s 2; x1 = -3s + 16 and x2 = 4s - 3 2(s - 2)(s + 2) (s - 2)(s + 2) C) s ± 2; x1 =

4s+ 3 -3s - 16 and x2 = 2(s - 2)(s + 2) (s - 2)(s + 2)

D) s ± 4; x1 = -3s - 16 and x2 = 4s + 3 49) Solve the system of equations using Cramer's Rule.

49)

sx1 - 7sx2 = 3

3x1 - 21sx2 = 5

4 9 - 5s A) s 1; x1 = and x2 = 21(s - 1)(s + 1) 21(s - 1)(s + 1)

B) s 1; x1 = 14 and x2 = 9 + 5s 3(s + 1) 21s(s + 1) C) s ± 1; x1 =

4 9 - 5s and x2 = 3(s + 1) 21s(s + 1)

D) s 0, 1; x1 =

4 9 - 5s and x2 = 3(s - 1) 21s(s - 1)

50) Solve the system of equations using Cramer's Rule.

50)

2x + y + z = a x + 2y + z = b x + y + 2z = c A) x = 4a + b + c , y = a + 4b + c , z = a + b + 4c 10 10 10

B) x = -4a + b + c , y = a - 4b + c , z = a + b - 4c 10

10

10

C) x = 4a + b - c , y = a + 4b - c , z = a - b + 4c 10

10

10

D) x = 3a - b - c , y = -a + 3b - c , z = -a - b + 3c 4

4

4

51) Calculate the area of the parallelogram induced by u = 1 , v = 2 .

2

A) -1

B) 7

C) -3

8

51)

3

D) 1

52) Calculate the area of the parallelogram form with the following points as vertices.

(0, 0), (5, 8), (13, 10), (8, 2) A) 53

B) 59

C) 108

D) 54

53) Calculate the area of the parallelogram form with the following points as vertices.

(-1, -2), (3, 6), (5, 0), (9, 8) A) 80

B) 40

421

1

1

1

114

0

0

-1

C) 39

B) 153

C) 105

0 t

Compute the area determined by the image vectors of 1 and 0 .

0 1 A) 0 B) 2 C) 2t

1

0

0

2

57) Compute the volume of the parallelepiped determined by -1 , 1 , and

0 1 0

C) 3

1

0

0

2

58) Let A = 1 0 0 , d = -1 , u = 1 , and v = 0 0 induced by

1

A d , A u and A v .

A) -7

-1 5 .

1

C) 7

D) -7

B) ii only D) all three sets are collinear

C) iii only

9

58)

D) 4

59) Determine which of the following sets of points are collinear. i) (2,3),(4,5),(6,8) ii) (-1,4)(2,1)(1,2) iii) (5,1),(3,1),(0,2) A) i only

56)

57)

-1 5 . Calculate the volume of the parallelepiped 1

B) 3

55)

D) t

1 1 0 0 1 0 1 56) Calculate the volume of the 4-dimensional parallelotope determined by , , , and 1 .

3 2 1 2 2 1 3 0 A) -2 B) 3 C) 2 D) 1

B) 7

54)

D) 109

55) Let A = 1 0 be the standard matrix of the stretch S: R2 R2 in the x direction by a factor of t.

A) 5

53)

D) 44

54) Let A = 1 4 1 , u = -1 , v = 2 , w = 5 . Calculate the volume of the parallelepiped induced

by A u , A v , and A w .

A) 12

52)

59)

60) Determine which of the following sets of points are not collinear. i) (2,3),(4,5),(6,7) ii) (1,4)(-2,1)(1,2) iii) (0,1),(3,1),(0,2) A) i only

60)

B) ii and iii only D) all three sets are notcollinear

C) all three sets are collinear

61) Determine whether the set of points (2,4)(-2,1)(1,2) is collinear, if they are not collinear, calculate

61)

the area of the triangle form with the points as vertices.

A) 5 B) The points are collinear

C) 5/2

D) 15

62) Determine whether the set of points (0,1)(-2,1)(1,1) is collinear, if they are not collinear, calculate the area of the triangle form with the points as vertices.

A) 1/2 B) 1

C) The points are collinear

D) 0

10

62)

Answer Key Testname: UNTITLED5

1) C 2) D 3) D 4) D 5) A 6) C 7) D 8) A 9) C 10) B 11) D 12) C 13) B 14) C 15) B 16) A 17) B 18) B 19) A 20) B 21) C 22) D 23) B 24) B 25) D 26) D 27) D 28) C 29) A 30) D 31) C 32) C 33) B 34) A 35) B 36) A 37) C 38) A 39) B 40) B 41) A 42) C

11

Answer Key Testname: UNTITLED5

43) B 44) B 45) B 46) B 47) A 48) C 49) D 50) D 51) D 52) D 53) B 54) B 55) D 56) C 57) B 58) C 59) B 60) B 61) C 62) C

12

Exam Name___________________________________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

1) For the given matrix and eigenvalue, find an eigenvector corresponding to the eigenvalue.

A=

32 -9 , 108 -31

=5

A) 1

C) 3

B) -3

3

D)

1

1

1 -3

2) For the given matrix and eigenvalue, find an eigenvector corresponding to the eigenvalue.

A = -24 -14 , 84 46

A)

B) 1

1 46

1 -4 -4 A= - 4 1 4 , 4 -4 -7

C)

0

1 -2

D) -2 1

1 0 -1

0 0 -4 2 -6 0 , -8 -16 2

A)

1 1 1 , 0 0 4

3)

= -3

B)

0 1 -1

C)

1 0 0 , 1 -1 1

D)

1 1 1 , 0 0 1

4) For the given matrix A, find a basis for the corresponding eigenspace for the given eigenvalue.

A=

2)

=4

3) For the given matrix A, find a basis for the corresponding eigenspace for the given eigenvalue.

A)

1)

= -4

B)

1 1 4

C)

1

1 -1 -4

D)

1 1 1 , 0 0 -4

4)

5) Determine if the vector is an eigenvector of a matrix. If it is, determine the corresponding

5)

eigenvalue.

4 -1 3 1 A = 2 1 3 and v = 2 2 -1 4 0

A) The eigenvalue is 1.

B) The eigenvalue is 0.

C) The eigenvalue is 2.

D) v is not an eigenvector.

6) Determine if the vector is an eigenvector of a matrix. If it is, determine the corresponding

6)

eigenvalue.

1 1 1 1 A = 1 1 1 and v = 1 1 1 1 1

A) The eigenvalue is 0.

B) v is not an eigenvector.

C) The eigenvalue is 3.

D) The eigenvalue is 2.

7) Determine if the vector is an eigenvector of a matrix. If it is, determine the corresponding

7)

eigenvalue.

1 1 1 0 A = 1 1 1 and v = 1 1 1 1 1

A) v is not an eigenvector.

B) The eigenvalue is 2.

C) The eigenvalue is 3.

D) The eigenvalue is 0.

8) Determine if the vector is an eigenvector of a matrix. If it is, determine the corresponding eigenvalue.

A = 1 1 and v = 0 11 1

A) The eigenvalue is 2.

B) The eigenvalue is 0.

C) The eigenvalue is 3.

D) v is not an eigenvector.

2

8)

9) Determine if the vector is an eigenvector of a matrix. If it is, determine the corresponding

9)

eigenvalue.

A = 1 1 and v = 1 11 1

A) The eigenvalue is 0.

B) The eigenvalue is 1.

C) The eigenvalue is 2.

D) v is not an eigenvector.

10) Determine if the vector is an eigenvector of a matrix. If it is, determine the corresponding

10)

eigenvalue.

A = 1 0 and v = 1 11 0

A) The eigenvalue is 3.

B) v is not an eigenvector.

C) The eigenvalue is 1.

D) The eigenvalue is 2.

11) Determine if the vector is an eigenvector of a matrix. If it is, determine the corresponding

11)

eigenvalue.

A = 1 0 and v = 1 11 1

A) The eigenvalue is 1.

B) The eigenvalue is 3.

C) v is not an eigenvector.

D) The eigenvalue is 2.

12) Find the eigenvalues of the given matrix. 0 -1 2

A) -2

B) 1, -2

13) Find the eigenvalues of the given matrix.

A) 5

12)

3

C) 1, 2

D) 1

85 16 -440 -83

B) -3, 5

13) C) 3

2 5 3 14) Find the characteristic equation of the given matrix. A = 0 4 -5 0 0 8 0 0 0

A) (2 - )(7 - )(8 - )(1 - ) = 0 C) (2 - )(4 - )(8 - ) = 0

D) 3, -5 1 8 7 2

B) (2 - )(5 - )(3 - )(1 - ) = 0 D) (2 - )2(4 - )(8 - ) = 0

3

14)

1 -7 4 9 15) Find the characteristic equation of the given matrix. A = 0 -5 7 -1 0 0 -7 5 0 0 0 6 A) (1 - )(-5 - )(-7 - )(6 - ) = 0 B) (1 - )(-7 - )(4 - )(9 - ) = 0

C) (9 - )(-1 - )(5 - )(6 - ) = 0

15)

D) (6 - )(5 - )(-1 - )(9 - ) = 0

16) The characteristic polynomial of a 5 × 5 matrix is 5 + 17 4 + 72 3. Find the eigenvalues and their

16)

multiplicities.

A) 0 (multiplicity 2), -9 (multiplicity 1), -8 (multiplicity 1)

B) 0 (multiplicity 3), 8 (multiplicity 1), 9 (multiplicity 1) C) 0 (multiplicity 1), 8 (multiplicity 1), 9 (multiplicity 1) D) 0 (multiplicity 3), -9 (multiplicity 1), -8 (multiplicity 1) 17) The characteristic polynomial of a 5 × 5 matrix is 5 - 13 4 - 9 3 + 405 2 . Find the eigenvalues

17)

and their multiplicities.

A) 0 (multiplicity 2), 9 (multiplicity 2), -5 (multiplicity 1)

B) 0 (multiplicity 1), 9 (multiplicity 3), -5 (multiplicity 1) C) 0 (multiplicity 2), -9 (multiplicity 2), -5 (multiplicity 1) D) 0 (multiplicity 2), -9 (multiplicity 2), 5 (multiplicity 1) 18) Given eigenvalue of a matrix A, determine the geometric and algebraic multiplicity of the

18)

eigenvalue.

1 1 1 A= 1 1 1 , =0 1 1 1 A) geometric multiplicity 2 and algebraic multiplicity 2

B) geometric multiplicity 1 and algebraic multiplicity 2 C) geometric multiplicity 2 and algebraic multiplicity 1 D) geometric multiplicity 1 and algebraic multiplicity 1 19) Given eigenvalue of a matrix A, determine the geometric and algebraic multiplicity of the eigenvalue.

3 1 1 A= 1 3 1 , =2 1 1 3 A) geometric multiplicity 1 and algebraic multiplicity 1

B) geometric multiplicity 2 and algebraic multiplicity 2 C) geometric multiplicity 1 and algebraic multiplicity 2 D) geometric multiplicity 2 and algebraic multiplicity 1

4

19)

20) Given eigenvalue of a matrix A, determine the geometric and algebraic multiplicity of the

20)

eigenvalue.

3 1 1 A= 1 3 1 , =5 1 1 3 A) geometric multiplicity 2 and algebraic multiplicity 2

B) geometric multiplicity 1 and algebraic multiplicity 1 C) geometric multiplicity 2 and algebraic multiplicity 1 D) geometric multiplicity 1 and algebraic multiplicity 2 21) Given eigenvalue of a matrix A, determine the geometric and algebraic multiplicity of the

21)

eigenvalue.

A= 1 0 , =1 1 1 A) geometric multiplicity 2 and algebraic multiplicity 1

B) geometric multiplicity 2 and algebraic multiplicity 2 C) geometric multiplicity 1 and algebraic multiplicity 2 D) geometric multiplicity 1 and algebraic multiplicity 1 22) Given eigenvalue of a matrix A, determine the geometric and algebraic multiplicity of the

22)

eigenvalue.

A = 1 0 where k 0, = 1 k 1 A) geometric multiplicity 2 and algebraic multiplicity 2

B) geometric multiplicity 1 and algebraic multiplicity 2 C) geometric multiplicity 1 and algebraic multiplicity 1 D) geometric multiplicity 2 and algebraic multiplicity 1 23) Given eigenvalue of a matrix A, determine the geometric and algebraic multiplicity of the

23)

eigenvalue.

A= 1 0 , =1 0 1 A) geometric multiplicity 1 and algebraic multiplicity 1

B) geometric multiplicity 1 and algebraic multiplicity 2 C) geometric multiplicity 2 and algebraic multiplicity 2 D) geometric multiplicity 2 and algebraic multiplicity 1 24) Let A be an n × n matrix such that A2 = A. Then the only possible eigen values of A are ________.

A) 1 and -1 B) -1 C) 0 and 1 D) 0 and -1

5

24)

25) Suppose that A is 3 × 3 matrix such that det(A) = 0, det(A - 2I) = 0 and det(A + I) = 0. Then what is the dimension of the solution space of A x = 0 ?

A) 3 B) 0

C) 1

D) 2

26) Suppose that A is 3 × 3 matrix such that det(A) = 0, det(A - 2I) = 0 and det(A + I) = 0. Then what is the dimension of the null space of A-3I? A) 0 B) 2

C) 1

B) 3

C) 0

B) a, b, and c

C) b and c

28) D) a and c

29) Which of the following statements are true for an n × n matrix A?

I) If A has no real eigen values, then A3 has no real eigen values.

II) If the eigen values of A are non zero, then A is invertible.

A) Neither I nor II B) I only C) II only

27)

D) 2

28) Let A= a 0 . Then the eigen values of A are ________. b c A) a and b

26)

D) 3

27) Suppose that A is 3 × 3 matrix such that det(A) = 0, det(A - 2I) = 0 and det(A + I) = 0. Then what is the rank of A? A) 1

25)

29) D) Both I and II

30) Determine whether P diagonalizes A for each of the following sets of matrices.

30)

I) A = 1 4 , P = 1 -4 3 2 1 3

II) A = 6 5 , P = 1 2 3 -7 1 1 A) Neither I nor II

B) Both I and II

C) I only

D) II only

31) Determine whether P diagonalizes A for each of the following sets of matrices.

31)

6 0 0 1 0 -1 I) A = 1 6 0 , P = 6 6 0 0 0 6 1 1 1

1 1 4 1 0 -1 0 -4 0 , P = -9 -4 0 -5 -1 -8 1 1 1 A) Neither I nor II B) Both I and II

II) A =

C) I only

6

D) II only

32) Determine whether P diagonalizes A for each of the following sets of matrices.

32)

3 1 1 1 -1 -1 I) A = 1 3 1 , P = 1 1 0 1 1 3 1 0 1

1 1 4 1 0 -1 0 -4 0 , P = -9 -4 0 -5 -1 -8 1 1 1 A) Neither I nor II B) Both I and II

II) A =

C) I only

D) II only

33) Determine whether P diagonalizes A for each of the following sets of matrices.

33)

6 0 0 1 0 -1 I) A = 1 6 0 , P = 6 6 0 0 0 6 1 1 1 II) A = 6 5 , P = 1 2 3 -7 1 1 A) Neither I nor II

B) Both I and II

C) I only

D) II only

34) Diagonalize the matrix A, if possible. That is, find an invertible matrix P and a diagonal matrix D

34)

1 1 4 such that A = PDP-1 , A = 0 -4 0 .

-5 -1 -8

1

0 -1 -4 0 0 0 ,D= 0 1 0 1 1 0 0 -3

B) P = -9 -4

1

1 -9 -1 -4 1 0 0 , D = 0 -4 0 1 -4 1 0 0 -3

D) P = -9 -4

A) P = 0 -4 1

1 1

C) P = -9 -4

1

0 -1 -4 0 -3 0 , D = 0 -4 0 1 1 0 -4 -3 0 -1 -4 0 0 0 , D = 0 -4 0 1 1 0 0 -3

35) Diagonalize the matrix A, if possible. That is, find an invertible matrix P and a diagonal matrix D 6 -9 -5 such that A = PDP-1 , A = -4 9 4 .

9 -37 4

1 0 -1

A) P = 0 3 0 , D = 1 1

1

1 0 -1

C) P = 5 3 0 , D = 1 1

1

-5 0 0 0 1 0 0 0 -2

B) P = 5 3 0 , D =

1 0 -1

-5 0 -2 0 -5 0 0 -5 -2

D) Not diagonizable

1 1

7

1

-5 0 0 0 -5 0 0 0 -2

35)

36) Diagonalize the matrix A, if possible. That is, find an invertible matrix P and a diagonal matrix D

36)

such that A = PDP-1 , A = 1 4 .

3 2

A) Not diagonalizable

B) P = 1 3 , D = 5 0

C) P = 1 -4 , D = 5 0

D) P = 1 -4 , D = 5 0

1 3

1 -4

0 -2

1 3

0 -2

0 2

37) Diagonalize the matrix A, if possible. That is, find an invertible matrix P and a diagonal matrix D such that A = PDP-1 , A =

A) Not diagonalizable

B) P = 1 -1 , D = 5 0

C) P = 1 2 , D = 5 0

D) P = -1 1 , D = 5 0

1 -1

1 2

03

1 -2

02

03

38) Diagonalize the matrix A, if possible. That is, find an invertible matrix P and a diagonal matrix D such that A = PDP-1 , A =

B) P = 1 -1 , D = 1 0

C) P = -1 1 , D = 1 0

D) Not diagonalizable

1 -2

0 1

1 2

0 3

0 1

39) Diagonalize the matrix A, if possible. That is, find an invertible matrix P and a diagonal matrix D 6 0 0 such that A = PDP-1 , A = 1 6 0 .

0 0 6

1 0 -1 6 0 1 0 ,D= 1 6 1 1 1 1 0 0 6

B) P = 6 6 0 , D = 0 6 0

1 6 1 6 0 0 = , D 0 6 1 0 6 0 0 0 6 -1 0 1

D) Not diagonalizable

A) P = 6 6

C) P =

38)

3 6 .

-5 -3

A) P = 1 2 , D = 1 0 1 -1

37)

7 2 .

-4 1

8

1 0 0

6 1 0

0 1 1

0 0 6

39)

40) Diagonalize the matrix A, if possible. That is, find an invertible matrix P and a diagonal matrix D

40)

-8 0 0 0 such that A = PDP-1 , A = 0 -8 0 0 .

1 -4 8 0 -1 2 0 8

16 -8 1 0 8 32 0 0 -8 A) P = ,D= 0 0 0 1 0 0 0 0 0 1 0

0 0 0 8 0 0 0 -8 0 0 0 -8

B) Not diagonalizable

16 32 0 0 -8 -8 0 0 , D = C) P = 1 0 1 0 0 1 0 1

0 0 0 8 0 0 0 -8 0 0 0 -8

16 32 0 0 -8 0 0 0 8 8 0 0 D) P = , D = 0 -8 0 0 1 0 1 0 0 0 8 0 0 1 0 1 0 0 0 8

8 0 0 0

41) Diagonalize the matrix A, if possible. That is, find an invertible matrix P and a diagonal matrix D

41)

-3 0 0 0 0 -3 0 0 . such that A = PDP-1 , A = - 12 3 -9 12 0 0 0 -3

A) Not diagonalizable

2 B) P = 0 1 0

0 -2 1 -3 0 0 0 2 1 0 , D = 0 -3 0 0 0 0 1 0 0 -3 0 0 1 0 0 0 0 -9

4 -2 1 0 -3 0 0 0 8 0 0 -2 C) P = , D = 0 -3 0 0 1 0 1 1 0 0 -9 0 0 1 1 0 0 0 0 -9

2 D) P = 0 -2 1

0 2 1 0

1 0 0 1

0 -3 0 0 0 0 , D = 0 -3 0 0 1 0 0 -3 0 0 0 0 0 -9

42) Diagonalize the matrix A, if possible. That is, find an invertible matrix P and a diagonal matrix D such that A = PDP-1 , A = 1 1 .

-1 1

A) P = -1 2 , D = 1 0

B) P = 1 2 , D = 1 0

C) Not diagonalizable over .

D) P = 1 -2 , D = 1 0

1 -1

1 -1

0 1

1 -1

9

0 1

0 1

42)

43) Diagonalize the matrix A, if possible. That is, find an invertible matrix P and a diagonal matrix D such that A = PDP-1 , A =

A) P = 1 2 , D = 1 0

B) P = -1 2 , D = 1 0

C) Not diagonalizable over .

D) P = 1 -2 , D = 1 0

1 -1

0 1

1 -1

1 -1

0 1

0 1

44) Let A be a diagonalizable matrix whose eigen values satisfy that 2 = + 1. Then A satisfies ________.

A) A2 = A + I

43)

0 1 .

-1 0

B) A2 = A + 1

C) A2 = A + 2

44)

D) A2 = A

45) Let A be a diagonalizable matrix. If 0 and 1 are the only eigenvalues of A, then ________.

A) A = I B) A2 = A + I C) A = 0 D) A2 = A

45)

46) Let A and B be n × n matrices such that P-1 A P = B for some invertible matrix P. Then which of

46)

the following statements are true?

I) det(A) = det(B) II) A and B have the same eigenvectors.

A) Neither I nor II B) Both I and II

C) II only

D) I only

47) Suppose that demographic studies show that each year about 6% of a city's population moves to

47)

the suburbs (and 94% stays in the city), while 4% of the suburban population moves to the city (and 96% remains in the suburbs). In the year 2000, 60.7% of the population of the region lived in the city and 39.3% lived in the suburbs. What is the distribution of the population in 2002? For simplicity, ignore other influences on the population such as births, deaths, and migration into and out of the city/suburban region.

A) 57.7% in the city and 42.3% in the suburbs

B) 56.8% in the city and 43.2% in the suburbs C) 55.7% in the city and 44.3% in the suburbs D) 58.6% in the city and 41.4% in the suburbs 48) Suppose that the weather in a certain city is either sunny, cloudy, or rainy on a given day, and

consider the following: 1) If it is sunny today there is a 70% chance it will be sunny tomorrow and a 30% chance that it will be cloudy.

2) If it is cloudy today there is a 40% chance it will be sunny tomorrow, a 40% chance that it will be cloudy, and a 20% chance that it will be rainy.

3) If it is rainy today there is a 40% chance it will be sunny tomorrow, a 30% chance that it will be cloudy, and a 30% chance that it will be rainy.

Suppose the predicted weather for Friday is 55% sunny, 35% cloudy, and 10% rainy. What are the chances that Sunday will be rainy?

A) 10% B) 11% C) 9.7% D) 8.7%

10

48)

49) Find the fixed-state vector for the Markov matrix P.

49)

P = 0.2 0.5 0.8 0.5 5 13

B) 5

A) 8

8

13

1 5

8 13

C) 4

D) 5

5

13

50) Find the fixed-state vector for the Markov matrix P. 0.9 P = 0.1 0

0.3 0.2 0.6 0.7 0.1 0.1 30 41

A)

29 9 1

50)

3 5

10

3

29 39 3

B) 41

C) 10

D) 13

1 41

1 10

1 39

51) Suppose that demographic studies show that each year about 6% of a city's population moves to

51)

52) Suppose that the weather in a certain city is either sunny, cloudy, or rainy on a given day, and

52)

the suburbs (and 94% stays in the city), while 2% of the suburban population moves to the city (and 98% remains in the suburbs). In the year 2000, 61.2% of the population of the region lived in the city and 38.8% lived in the suburbs. What percentage of the population of the region would eventually live in the city if the migration probabilities were to remain constant over many years?

For simplicity, ignore other influences on the population such as births, deaths, and migration into and out of the region.

A) 37.5% B) 75.0% C) 50% D) 25%

consider the following: 1) If it is sunny today there is a 70% chance it will be sunny tomorrow and a 30% chance that it will be cloudy.

2) If it is cloudy today there is a 40% chance it will be sunny tomorrow, a 40% chance that it will be cloudy, and a 20% chance that it will be rainy.

3) If it is rainy today there is a 40% chance it will be sunny tomorrow, a 30% chance that it will be cloudy, and a 30% chance that it will be rainy.

In the long run, how likely is it that the weather will be rainy on a given day?

A) 9.5% B) 10.6% C) 12.1%

11

D) 11.3%

53) Find a formula for Ak, given that A = PDP-1 , where P and D are given below.

A=

53)

5 3 ,P= 3 1 ,D= 7 0 -2 10 21 0 8

A)

7k

k

0

8k

0

k

B) 3 · 7 - 2 · 8

k

k

k

2 · 7k - 2 · 8k

C) 3 · 7 - 2 · 8

k

2 · 7k + 2 · 8k

3 · 8k + 3 · 7k 3 · 8k - 2 · 7k

D) 3 · 7 + 2 · 8

2 · 7k + 2 · 8k

3 · 8k - 3 · 7k 3 · 8k - 2 · 7k 3 · 8k + 3 · 7k 3 · 8k + 2 · 7k

54) Find a formula for Ak, given that A = PDP-1 , where P and D are given below.

54)

A= 0 1 ,P= 1 1 ,D= 2 0 2 1 2 -1 0 -1

A)

C)

2k 0

0

B)

(-1)k 2(-1)k + 2k 3

(-1)k+1 + 2k 3

2(-1)k+1 + 2k+1 3

(-1)k+2 + 2k+1 3

D)

55) Use diagonalization to calculate A10 where A = A)

1

2 10

0

B)

310

1

-2 10

0

2(-1)k + 2k 3

(-1)k+1 + 2k 3

2(-1)k+1 + 2k+1 3

(-1)k+1 + 2k+1 3

2(-1)k + 2k 3

(-1)k+1 + 2k 3

2(-1)k+1 + 2k+1 3

(-1)k+2 + 2k+2 3

1 0 .

-2 3

55)

C)

310

1 -1

0

0

3

1

0

1-3 10

310

56) Use diagonalization to calculate A10 where A = 0 2 1 .

A)

1 0 0

1 1024 0

1 -1023 1024 0 0

C) 0

0 1 59049

0

1 -1025 1024 0 0

-28499 58025 59049

1

1023

0

0

28501 58025 59049

D) 0 1024

12

0

59049

56)

B) 0

-28501 58025 59049

D) 1 -59048

57) Solve the initial value problem. x = Ax, x(0) =

57)

3 , where A = -4 -3.125 .

-4 3.2 8

A) x(t) =

2 sin 5t - 3 cos 5t e-4t -3.2 cos 5t - 4.8 sin 5t

B) x(t) = -2 sin 5t + 3 cos 5t e-4t

C) x(t) =

2 sin 5t + -3 cos 5t e4t -3.2 cos 5t - 4.8 sin 5t

D) x(t) = -2 sin 5t + 3 cos 5t e4t

3.2 cos 5t + 4.8 sin 5t

3.2 cos 5t + 4.8 sin 5t

58) Find the general solution of the system of linear differential equations.

58)

x = Ax, where A = 3 2 .

4 -4

A) x(t) = ae-5t -1 + be4t 2 4

B) x(t) = ae5t -1 + be4t 2 4

1

C) x(t) = ae-5t -1 + be-4t 2 4

1

D) x(t) = ae-5t 1 + be4t 2

1

4

13

1

Answer Key Testname: UNTITLED6

1) A 2) C 3) D 4) B 5) C 6) C 7) A 8) D 9) C 10) B 11) C 12) C 13) B 14) D 15) A 16) D 17) A 18) A 19) B 20) B 21) C 22) B 23) C 24) C 25) C 26) A 27) D 28) D 29) C 30) C 31) D 32) B 33) A 34) D 35) D 36) C 37) D 38) D 39) D 40) D 41) D 42) C

14

Answer Key Testname: UNTITLED6

43) C 44) A 45) D 46) D 47) B 48) C 49) A 50) D 51) D 52) A 53) B 54) C 55) C 56) C 57) B 58) A

15

Exam Name___________________________________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

1) Determine if the given set of vectors in Rn is orthogonal and, if so, normalize to make an

1)

orthonormal set.

3 -3 3 6 , 0 , -3 3 3 3

A) B)

1 1 -1 1 1 1 2 , 0 , -1 6 3 2 1 1 1 1 1 -1 1 1 1 2 , 0 , -1 6 2 3 1 1 1

1 1 -1 C) 1 2 , 1 0 , 1 -1 6 2 3 1

1

1

D) The set is not an orthogonal set of vectors.

2) Determine if the given set of vectors in Rn is orthogonal and, if so, normalize to make an orthonormal set.

1 , 6 -2 3

A) The set is not an orthogonal set of vectors.

B) 1 1 , 1 2 5 -2

15 1

C)

1 1 , 1 2 5 -2 5 1

D)

1 1 , 1 2 5 -2 5 1

1

2)

3) Determine if the given set of vectors in Rn is orthogonal and, if so, normalize to make an

3)

orthonormal set.

-2 -2 3 0 , 6 , 5 1 -4 6

A)

3 -2 -1 1 1 , , 0 3 5 14 70 1 6 -2

1 5

B) The set is not an orthogonal set of vectors.

3 -2 -1 C) 1 0 , 1 3 , 1 5 5 14 70 1

D)

1 3

6

-2

3 -2 -1 1 1 0 , 3 , 5 14 14 1 6 -2

4) Determine if the given set of vectors in Rn is orthogonal and, if so, normalize to make an orthonormal set.

0 -1 1 , 0 , 0 1 0 -1

A)

1 2

1 1 1 , -1 1 1 1 -1

0 -1 1 ,1 0 ,1 0 2 1 2 -1 0

1 1 1 , 1 -1 1 2 1 -1 1

B) The set is not an orthogonal set of vectors.

C)

1 2

D)

1 2

0 -1 1 , 1 0 ,1 0 2 1 2 -1 0

0 -1 1 ,1 0 ,1 0 2 1 2 -1 0

1 1 1 , 1 -1 1 2 1 -1 1

1 1 1 , 1 -1 1 2 1 -1 1

2

4)

5) Determine if the given set of vectors in Rn is orthogonal and, if so, normalize to make an

5)

orthonormal set.

2 -1 , 1

1 1 2 , -1 1 1

2 A) 1 -1 , 1 6 6

B)

1 1 1 , 2 -1 3 1 1 1 2 1 1 1 1 1 2 , -1 , -1 6 6 3 1 1 1

C) The set is not an orthogonal set of vectors.

2 D) 1 -1 , 1 6 6 1

1 1 1 2 , -1 6 1 1

6) Determine if the given set of vectors in Rn is orthogonal and, if so, normalize to make an orthonormal set.

1 0 , 1 0

1 1 , 1 1

1 3 3 1

1 1 1 1 0 1 1 1 3 A) , , 2 1 4 1 20 3 0 1 1 B) The set is not an orthogonal set of vectors.

C)

1 1 0 1 , 2 1 2 0

D)

1 1 0 1 , 2 1 2 0

1 1 , 1 1 20 1

1 3 3 1

1 1 , 1 1 20 1

1 3 3 1

3

6)

7) Determine if the given set of vectors in Rn is orthogonal and, if so, normalize to make an

7)

orthonormal set.

-2 1 1 2 1 , 0 , -3 , 1 1 1 -1 -1 0 1 0 -1

A)

1 6

-2 1 , 1 1 3 -0

1 1 2 0 , 1 -3 , 1 1 1 11 -1 7 -1 -1 1 0

B) The set is not an orthogonal set of vectors.

C)

1 6

-2 1 ,1 1 3 -0

1 1 2 1 1 0 , 1 -3 , 1 11 -1 7 -1 -1 1 0

D)

1 2

-2 1 ,1 1 3 -0

1 1 2 1 1 0 , 1 -3 , 1 5 -1 5 -1 -1 1 0

8) Determine if the given set of vectors in Rn is orthogonal and, if so, normalize to make an orthonormal set.

1 , 2 -2 1

A) The set is not an orthogonal set of vectors.

B)

1 1 , 1 2 5 -2 5 1

C) 1 1 , 1 2 5 -2

D)

15 1

1 , 2 -2 1

4

8)

9) Determine if the given set of vectors in Rn is orthogonal and, if so, normalize to make an

9)

orthonormal set.

1 , -2 -2 1

A)

1 1 , 1 -2 5 -2 5 1

B) The set is not an orthogonal set of vectors.

C) 1 1 , 1 -2 5 -2

D)

15

1

1 , -2 1 -2

10) Determine if the given set of vectors in Rn is orthogonal and, if so, normalize to make an orthonormal set.

1 0 1 0 1 -3 , 3 1 11 -1 1 1

A) The set of vectors are orthonormal.

B)

1 6

-2 1 , 1 1 3 -0

1 1 2 0 , 1 -3 , 1 1 1 11 -1 7 -1 -1 1 0

1 1 2 -2 1 0 1 -3 1 1 1 C) , , , 1 3 1 5 -1 5 -1 -0 -1 1 0 D) The set is not an orthogonal set of vectors.

1 2

5

10)

11) Determine if the given set of vectors in Rn is orthogonal and, if so, normalize to make an

11)

orthonormal set.

1 -1 1 1 0 , 0 , 0 1 -1 0

1 0 1 1 1

1 1 -1 1 1 0 A) 1 0 , 1 0 , 1 1 3 3 3 0 1 1 0 1 -1 B) The set is not an orthogonal set of vectors.

1 -1 1 1 1 1 1 0 , 0 , 2 3 3 0 1 -1 0

C)

1 0 1 1 1

D) The set of vectors are orthonormal.

12) Given B is an orthonormal basis, then write the vector x as a linear combination of the vectors in B.

B=

A)

1 2

-1 0 1 1 1 , 1 0 , 1 1 , 1 -1 0 2 1 2 1 2 1 0 1 -1 -1

2 2 5 -1

B)

1 2 and x = 3 4

- 2 2 -1 5

C)

6

- 2 2 5 -1

D)

- 2 - 2 5 -1

12)

13) Given B is an orthonormal basis, then write the vector x as a linear combination of the vectors in

13)

B.

B=

1 2

-1 0 1 1 , 0 ,1 0 2 1 2 0 -1

1 1 1 1 , -1 1 2 1 1 -1

0 -3 2

A)

1 2

3 and x = 1 0 1 0 3 2

1 -3 2 1 2

B)

5 2

1 2

C)

0 -3 2 5 2

D)

5 2

5 2

1 2

14) Given B is an orthonormal basis, then write the vector x as a linear combination of the vectors in

14)

B.

B=

1 1 , 1 2 5 -2 5 1 1 5

A)

2 5

and x = 1 0 -

B) 1

C)

0

1 5 2 5

D)

-

1 5

-

2 5

15) Given B is an orthonormal basis, then write the vector x as a linear combination of the vectors in B.

B=

1 1 , 1 2 5 -2 5 1

and x = 0 1

2 5

A)

1 5

B)

-

1 5

-

2 5

-

C) 0 1

7

D) -

2 5 1 5

15)

16) Given B is an orthonormal basis, then write the vector x as a linear combination of the vectors in

16)

B.

B=

-1 1 1 1 1 1 , , 2 0 -1 6 2 3 1 1 1 5 0 0

A)

B)

1 and x = 2 .

1

6 0 0

C)

2 0 0

2

D) 0 0

17) Given B is an orthonormal basis, then write the vector x as a linear combination of the vectors in

17)

B.

B=

-1 1 1 1 1 1 1 0 , 0 , 2 3 3 0 1 -1 0

A)

3 3 2 3 2 0 0

18) Let B =

1 2

y B=

2 5 and x = 1 .

3 -2

B)

3 3 2 3 2

-1 0 1 1 1 , 1 0 , 1 1 , 1 -1 0 2 1 2 1 2 1 0 1 -1 -1

2 2 . Determine 3 1

A) 30, 18

1 0 1 1 1

C)

, and x , y

3 3 0 2 0 0

D)

3 3 0 2

- 2 2 and 4 such that x B = 5 -1

x 2 and x y .

B) 6, 14

C) 6, 18

8

D) 30, 14

18)

19) Decide whether the matrix A is orthogonal. If A is not orthogonal, indicate how the columns of A

19)

fail to form an orthonormal set.

1 2

1 2

1 2

0

1 2

0

1 2

1 2

1 2

0

1 2

0

A=

-

-

1 2 -

1 2 1 2

-

1 2

A) A is not orthogonal. The columns are not unit vectors.

B) A is not orthogonal. The second and third columns are not orthogonal.

C) A is orthogonal.

D) A is not orthogonal. The first and third columns are not orthogonal.

20) Decide whether the matrix A is orthogonal. If A is not orthogonal, indicate how the columns of A

20)

fail to form an orthonormal set.

1 3

2 3

2 3

2 A= 3

2 3

1 3

2 3

1 3

2 3

A) A is not orthogonal. The columns are not orthogonal.

B) A is not orthogonal. The columns are not orthogonal and not unit vectors.

C) A is not orthogonal. The columns are not unit vectors.

D) A is orthogonal.

21) Decide whether the matrix A is orthogonal. If A is not orthogonal, indicate how the columns of A fail to form an orthonormal set.

A=

-2 -1 0 3 1 -2

3 5 6

A) A is not orthogonal. The columns are not orthogonal.

B) A is not orthogonal. The columns are not unit vectors.

C) A is orthogonal.

D) A is not orthogonal. The columns are not orthogonal and not unit vectors.

9

21)

22) Decide whether the matrix A is orthogonal. If A is not orthogonal, indicate how the columns of A

22)

fail to form an orthonormal set.

A = cos sin

-sin cos

A) A is not orthogonal.The columns are not orthogonal.

B) A is not orthogonal. The columns are not unit vectors.

C) A is orthogonal.

D) A is not orthogonal. The columns are not orthogonal and not unit vectors.

23) Decide whether the matrix A is orthogonal. If A is not orthogonal, indicate how the columns of A

23)

fail to form an orthonormal set.

A= 1 2

-2 1

A) A is not orthogonal. The columns are not orthogonal and not unit vectors.

B) A is orthogonal.

C) A is not orthogonal. The columns are not orthogonal.

D) A is not orthogonal. The columns are not unit vectors.

24) Decide whether the matrix A is orthogonal. If A is not orthogonal, indicate how the columns of A

24)

fail to form an orthonormal set.

A= a b

-b , where a and b are non-zero real numbers. a

A) A is not orthogonal. The columns are not orthogonal and not unit vectors.

B) A is not orthogonal. The columns are not unit vectors.

C) A is not orthogonal. The columns are not orthogonal.

D) A is orthogonal.

25) Let W be the subspace spanned by the u's. Write y as the sum of a vector in W and a vector orthogonal to W.

-1 12 2 y = 14 , u1 = 2 , u2 = 3 25 4 -1

1

11

17

8

1

-11 7 -8

A) y = 21 + -7

C) y = 21 + 17

1

13

17

42

2

10

34

-9

B) y = 21 + 35

D) y = 42 + -28

10

25)

26) Let W be the subspace spanned by the u's. Write y as the sum of a vector in W and a vector

26)

orthogonal to W. y=

21 1 2 = = , u , u 3 0 1 1 2 9 2 -1

A) y =

15 6 0 + 3 6 -15 20

C) y = 7 + 8

20

1

8

1

B) y = 7 + -4

-1 4 -1

20

41

8

17

D) y = 7 + 10

27) Find the vector closest to y in the subspace W spanned by u1 and u2 .

27)

10 2 -1 y = 20 , u1 = 2 , u2 = 3 -1 33 4

11

A) 23

B)

5

1

-8 96 95

C) -27

-1

D) 27 25

-25

28) Find the vector closest to y in the subspace W spanned by u1 and u2 . y=

28)

15 1 2 1 , u1 = 0 , u 2 = 1 -1 7 2

A)

20 9 16

B)

13 4 3

C)

11

-14 -5 -6

D)

14 5 6

29) Determine the projection of v onto the subspace spanned by the given orthogonal set.

1 2 v = -1 0 0

=

1 -1 1 1 1 0 , 0 0 , 1 0 1 1 0 1 -1

4 3

2 3

4 3

A)

-

B)

0 1 3 -1

-

4 3

C)

0 1 3

2 3

2 3

4 3

4 3

D)

0 1 3 -1

-1

0 1 3 -1

30) Determine the projection of v onto the subspace spanned by the given orthogonal set.

1 v= 2 -1 -2

A)

=

0 -1 1 , 0 , 0 1 -1 0

1 2 -1 -2

31) Let W = Span

29)

1 0 1 0

B)

0 -1 1 , 0 0 1 -1 0

A) Span

1 0 , -1 0

0 1 0 1

C) Span

1 0 , 1 0

0 1 0 1

30)

-1 2 1 -2

C)

-1 2 -1 -2

D)

. Find the orthogonal complement of W, W in 4 .

12

B) Span

1 0 , 1 0

1 1 1 1

D) Span

1 1 1 , -1 1 1 1 -1

-1 2 -1 2

31)

0 2 1 , -1 0 1 1 3

32) Let W = Span

A) Span

1 0 , 1 0

C) Span

-2 -1 -1 , 0 0 2 1 1

A) Span

3 0 1 1

1 -1 1 , 1 1 3

33) Let W = Span 1 2 1

B) Span

1 0 , 1 0

D) Span

1 3 0 , 1 -1 0 0 1

3 1 0 1

1 -2 1

C) Span

-1 2 -1

2 -1 1 , 0 0 1

B) Span

1 -2 1 , 0 0 1

C) Span

2 1 1 , 0 0 1

D) Span

-2 -1 1 , 0 0 1

3 1

A) Span

1 3

33)

D) Span

1 2 -1

. Find the orthogonal complement of W, W in 3 .

A) Span

35) Let W = Span

32)

. Find the orthogonal complement of W, W in 3 .

B) Span

1 2 1

34) Let W = Span

. Find the orthogonal complement of W, W in 4 .

34)

. Find the orthogonal complement of W, W in 2 .

B) Span

1 -3

C) Span

13

-1 -3

35) D) Span

3 1

0 -1 1 , 0 , 0 1 -1 0

36) Let W = Span

A) Span

0 0 0 -1

1 0 1 0

. Find the orthogonal complement of W, W in 4 .

B) Span

0 1 0 1

C) Span

0 0 0 1

36)

D) Span

1 -1 1 -1

37) The given set is a basis for a subspace W. Use the Gram-Schmidt process to produce an

37)

orthogonal basis for W.

4 10 Let x1 = -2 , x2 = -30 0 2

-6

10

0

2

4

-10

0

-2

B) -2 , -30

A) -2 , -30

4

30

0

2

C) -2 , -40

4

-10

0

2

D) -2 , -20

38) The given set is a basis for a subspace W. Use the Gram-Schmidt process to produce an orthogonal basis for W.

0 1 1 Let x1 = , x2 = 1 , x3 = -1 -1 1 -1

A)

0 3 18 1 , 4 , 4 -1 -4 19 1 -2 13

C)

0 1 1 , 1 , -1 -1 1 -1

1 0 1 1

1 0 1 1

14

B)

0 3 14 1 , 2 , 2 9 -1 -2 1 -4 7

D)

0 1 1 , 0 , 0 -1 1 -2

6 0 1 3

38)

39) The given set is a basis for a subspace W. Use the Gram-Schmidt process to produce an

39)

orthonormal basis for W. 1 1 1 , 0 1 , 1 0 0 1

A)

1 0 1 0 , 1 , 0 0 0 1

B)

1 1 0 0 , 1 , 0 0 0 1

C)

1 0 0 0 , 1 , 0 0 1 1

D)

1 0 0 0 , 1 , 0 0 0 1

40) The given set is a basis for a subspace W. Use the Gram-Schmidt process to produce an orthonormal basis for W. 1 0 0 2 , 1 , 0 3 0 1

A)

1 -1 -3 2 , 5 , 0 3 -3 -1

B)

1 -1 -3 2 , 5 , 0 3 -3 1

C)

1 1 -3 2 , 5 , 0 3 -3 -1

D)

1 -1 -3 2 , 5 , 1 3 -3 1

15

40)

41) The given set is a basis for a subspace W. Use the Gram-Schmidt process to produce an

41)

orthonormal basis for W. 1 1 1 0 , 1 , 3 1 1 3 0 1 1

0 1 2

1 2

A)

0 , 1 2

0 1 2

1 2 0 , 1 2

0 , 1 2

0

1 2

1 2

1 2

0 , 1 2

0

C)

-

B)

1 2 1 2 -

0

1 2

1 2

1 2

D)

1 2 -

0 , 1 2

1 2

0 , 1 2 0

0 1 2

1 2 -

1 2

0 , 1 1 2 2 1 2

0 1 2

1 2 1 2

0 , 1 1 2 2 1 2

42) Find the equation y = 0 + 1 x of the least-squares line that best fits the given data points.

42)

Data points: (5, -3), (2, 2), (4, 3), (5, -1) 1 x= 1 1 1

-3 5 2 ,y= 2 4 3 5 -1

A) y = 19 - 8 x 2

3

B) y = 67 + 0x

C) y = 67 - 2x

12

12

16

D) y = 67 - 4 x 12

3

43) Find the equation y = 0 + 1 x of the least-squares line that best fits the given data points.

43)

Data points: (2, 1), (3, 2), (7, 3), (8, 4) 1 2 x= 1 3 ,y= 1 7 1 8

1 2 3 4

A) y = 5 + 4 x 13

13

B) y = - 5 + 11 x 13

C) y = 5 + 11 x

26

13

26

D) y = - 3 + 11 x 26

44) Find the equation y = at2 + bt of the least-squares line that best fits the given data points.

Data points: (-1,4), (0,1), (1,1) A) y = 5 t2 + 3 t B) y = - 5 t2 + 3 t 2 2 2 2

C) y = 5 t2 - 3 t 2

2

44)

D) y = - 5 t2 - 3 t 2

2

45) Given: The data points (-3, 2), (-2, 5), (0, 5), (2, 2), (3, 7).

45)

Suppose the errors in measuring the y-values of the last two data points are greater than for the other points. Weight these data points half as much as the rest of the data.

1 -3 1 -2 X= 1 0 , 1 2 1 3

=

2 5 1 ,y= 5 2 2 7

A) y = 0.9 + 1.54x

B) y = 4.6 + 0.45x

C) y = 0.45 + 4.6x

D) y = 4.9 + 1.54x

46) Given: The data points (-2, 2), (-1, 5), (0, 5), (1, 2), (2, 2).

46)

Suppose the errors in measuring the y-values of the last two data points are greater than for the other points. Weight these data points twice as much as the rest of the data.

1 -2 1 -1 X= 1 0 , 1 1 1 2

=

2 5 1 ,y= 5 2 2 2

A) y = 2.8 - 0.55x

B) y = 2.9 - 0.45x

C) y = 5.8 - 0.90x

17

D) y = 2.6 - 0.53x

47) Given A and b, determine the least-squares error in the least-squares solution of Ax = b.

4 3 A= 2 1 ,b= 3 2

47)

4 2 1

A) 91.929925

B) 1.63299316

C) 3.82970843

D) 260.495468

48) Find a least-squares solution x of the inconsistent system A x = b, that minimizes A x - b where

48)

1 2 3 A= 3 4 ,b= 2 .

5 9 1

57 7016

A)

-

37 84

B)

21 7016

-

C)

5 21

-

3 4

-

1 4

19 18

D)

-

7 18

49) Find a least-squares solution x of the inconsistent system A x = b, that minimizes A x - b where 1 1 A= 1 1 1 1

1 1 0 0 0 0

0 0 1 1 0 0

0 0 0 ,b= 0 1 1

7 8 0 .

2 4 1

5 2

A)

C)

5 2

-1 5 + x -1 4 3 1 2 1

B)

5 7 2

0

0

5 2

5 4

4 3 2

-1 + x4 1 1 1

D)

0

5 3 2 0

18

-1 + x4 1 1 1

-1 + x4 1 1 0

49)

50) For p(t), q(t) P2, define p, q = p(t0 ) q(t0 ) + p( t1 ) q( t1 ) + p(t2 )q(t2), where t0 , t1 , t2 are distinct real

50)

numbers. Compute the length of the given vector.

1 p(t) = 10t2 and q(t) = t - 1, where t0 = 0, t1 = , t2 = 1 2

A) p = 5 17; q =

5 4

B) p =

= 10

5 4

C) p = 0.5 70; q =

5 4

D) p = 10 17; q =

5 4

2

5 ; 4

q

51) For p(t), q(t) P2, define p, q = p(t0 ) q(t0 ) + p( t1 ) q( t1 ) + p(t2 )q(t2), where t0 , t1 , t2 are distinct real

51)

numbers. Compute the length of the given vector. p(t) = 4t + 7 and q(t) = 7t - 5, where t0 = 0, t1 = 1, t2 = 2

A) p = 395; q = 340 C) p = -297; q = -60

B) p = 353; q = 390 D) p = 395; q = 110

52) For p(t), q(t) P2, define p, q = p(0) q(0) + p(1) q(1) + p(2)q(2). Calculate 1 - 2x - x2 .

A) 4 6 B) 3 6 C) 6 D) 2 6

52)

53) Find the norm of A= 1 2 in M(2,2) under the inner product A, B = tr(BTA).

53)

01

A) 4 6

B) 3 6

C) 2 6

D) 6

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

54) With the given positive numbers, show that vectors u = (u1, u2 ) and v = (v1 , v2 ) define an

54)

inner product on R2 using the 4 axioms.

Set u, v = 2u1 v1 + 2u2 v2 .

55) Let t0 , ...., tn be distinct real numbers. For p and q in n, define p, q = p(t0 )q(t0 ) +

55)

p(t1 )q(t1 ) + ... + p(tn )q(tn ). Show that p,q defines an inner product on Pn .

56) For f, g in C[a, b], set f, g =

b

f(t) g(t) dt.

56)

a Show that f, g defines an inner product on C[a, b].

19

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

57) Determine which of the following , defines an inner product on P2.

57)

I) For p(t), q(t) P2 , define p, q = p(0) q(0) + p(1) q( 1).

II) For p(t), q(t) P2 , define p, q = p(0) q(0) + p(1) q( 1) + p(2)q(2).

A) Both I and II

B) II only

C) I only

D) Neither I nor II

58) Let V be in 4 with inner product defined as

58)

<p,q> = p(t0 )q(t0 ) + p(t1 )q(t1 ) + p(t2 )q(t2 ) + p(t3 )q(t3 ) + p(t4 )q(t4 ),

involving evaluation of polynomials at -3, -1, 0, 1, and 3, and view 2 as subspace of V. Produce an orthogonal basis for P2 by applying the Gram-Schmidt process to the polynomials 1, t, and t2 .

A) 1, t, t2 - 4 B) 1, t, t2 - 5 C) 1, t, t2 + 4 D) 1, t, t2 + 16 3 5

59) Let V be in 4 with inner product defined as

59)

<p,q> = p(t0 )q(t0 ) + p(t1 )q(t1 ) + p(t2 )q(t2 ) + p(t3 )q(t3 ) + p(t4 )q(t4 ),

involving evaluation of polynomials at -6, -3, 0, 3, and 6, and view 2 as subspace of V. Produce an orthogonal basis for P2 by applying the Gram-Schmidt process to the polynomials 1, t, and t2 .

A) 1, t, t2 +18 B) 1, t, t2 -18 C) 1, t, t2 - 72 D) 1, t, t2 - 18 5 5

60) Define the inner product x , y = 2x1y1 + x2 y2 + 3x3y3 on 3. Use the Gram-Schmidt procedure to produce an orthogonal basis for

A)

1 -1 0 1 , 2 , 0 0 0 1

C)

1 -1 -1 1 , 1 , 1 0 0 -2

=

1 -1 -1 1 , 1 , 1 0 0 -2

60)

.

B)

1 1 0 1 , 2 , 0 0 0 1

D)

1 -1 0 1 , -2 , 0 0 0 1

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

61) Let V be the space C[0, 1] and let W be the subspace spanned by the polynomials

61)

p1 (t) = 1, p2 (t) = t - 1, and p3 (t) = 12t2 . Use the Gram-Schmidt process to find an orthogonal basis for W.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

20

62) Let C[0, ] have the inner product f, g =

f(t)g(t) dt, and let m and n be unequal positive

0 integers. Prove that cos(mt) and cos(nt) are orthogonal.

A) cos(mt), cos(nt) =

cos(mt)cos(nt)dt 0

[cos(mt + nt) + cos(mt - nt)]dt 0 sin(mt - nt) sin(mt + nt) from [0, ] = + m+n m-n =

= 1.

B) cos(mt), cos(nt) =

cos(mt)cos(nt)dt 0

=

1 2

[cos(mt + nt) + cos(mt - nt)]dt

0 1 sin(mt + nt) sin(mt - nt) from [0, ] = + 2 m+n m-n = 0.

C) cos(mt), cos(nt) =

cos(mt)cos(nt)dt 0 1 2

[cos(mt + nt) - cos(mt - nt)]dt 0 1 sin(mt + nt) sin(mt - nt) from [0, ] = 2 m-n m+ n =

= 0.

D) cos(mt), cos(nt) =

cos(mt)cos(nt)dt 0

=

1 2

[cos(mt + nt) + cos(mt - nt)]dt

0 1 sin(mt + nt) sin(mt - nt) from [0, ] = + 2 m-n m+n = 1.

21

62)

63) Find the nth-order Fourier approximation to the function f(t) = 4t on the interval [0, 2 ].

A)

8 - cos(t) - cos(2t) - cos(3t) - ... - cos(nt) n

B) 4 - 8sin(t) - 4sin(2t) - 8 sin(3t) - ... - 8 sin(nt) 3

n

C) 4 - 8sin(t) - 4sin(2t) - 4 sin(3t) - ... - 4 sin(nt) 3

n

D) 4 - 8cos(t) - 4sin(2t) - 8 cos(3t) - ... - 8 cos(nt) 3

n

22

63)

Answer Key Testname: UNTITLED7

1) B 2) D 3) A 4) C 5) C 6) B 7) B 8) B 9) B 10) A 11) C 12) C 13) D 14) A 15) D 16) B 17) B 18) D 19) C 20) A 21) B 22) C 23) D 24) B 25) A 26) B 27) D 28) D 29) D 30) A 31) C 32) C 33) B 34) D 35) B 36) B 37) D 38) B 39) D 40) B 41) C 42) D

23

Answer Key Testname: UNTITLED7

43) C 44) C 45) B 46) B 47) B 48) D 49) C 50) A 51) D 52) B 53) D 54) Axiom 1: u, v = 2u1 v1 + 2u2 v2 = 2v1u1 + 2v2 u2 = v, u Axiom 2: If w = (w1 , w2 ), then u + v, w = 2(u1 + v1 )w1 + 2(u2 + v2 )w2 = 2u1 w1 + 2u2 w2 + 2v1 w1 + 2v2 w2 = u, w + v, w Axiom 3: cu, v = 2c(u1 )v1 + 2(cu2 )v2 = c(2u1 v1 + 2u2 v2 ) = c u, v Axiom 4:

u, u = 2u1 2 + 2u2 2 0, and 2u1 2 + 2u2 2 = 0 only if u1 = u2 = 0. Also, 0, 0 = 0.

55) Axioms 1-3 are readily checked. For Axiom 4, note that p, p = [p(t0)]2 + [p(t1)]2 +... + [p(tn)]2 0.

Also, 0, 0 = 0.

If p, p = 0, then p must vanish at n + 1 points: t0 , ...., tn . This is possible only if p is the zero polynomial, because the degree of p is less than n + 1.

56) Answers will vary.

Inner product Axioms 1-3 follow from elementary properties of definite integrals. b Axiom 4: f, f = [f(t)]2 dt 0. a The function [f(t)]2 is continuous and nonnegative on [a, b]. If the definite integral of [f(t)]2 is zero, then [f(t)]2 must be identically zero on [a, b]. Thus, f, f = 0 implies that f is the zero function on [a, b].

57) B 58) A 59) B 60) A 61) As a function, q3 (t) = 12t2 - 48t + 20. The orthogonal basis for the subspace W is {q1, q2, q3}.

62) B 63) B

24

Exam Name___________________________________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

1) Determine which of the following matrices are symmetric.

5 2 4 A = 1 2 , B = 2 -5 -2 5 0 4 -2 0 A) None of them

B) A and B

1)

C) B only

D) A only

2) Determine which of the following matrices are symmetric.

12 1 5 2 4 A = 2 0 3 , B = 2 -5 -2 1 3 -1 4 -2 0 A) None of them B) A and B

2)

C) B only

D) A only

3) Which of the following statements are true for an n×n matrix A? I) ATA is symmetric.

II) If A is symmetric, then A is diagonalizable.

A) None B) Both

3)

C) I only

D) II only

4) Which of the following statements are true for an n × n matrix A?

4)

I) AT + A is symmetric.

II) If A is symmetric, then all the eigen values of A are real.

A) None B) Both C) I only

D) II only

5) Orthogonally diagonalize the matrix, giving an orthogonal matrix P and a diagonal matrix D. 2 2 2 5

A) P = 1 -2 , D = 6 0 2 1

C) P =

B) P = 1 -2 , D = 7 0

0 1

2 1

0 2

D) P = 1/ 5 -2/ 5 , D = 6 0

1/ 5 -2/ 5 , D = 1 0 0 6 -2/ 5 1/ 5

2/ 5

1

1/ 5

0 1

5)

6) Orthogonally diagonalize the matrix, giving an orthogonal matrix P and a diagonal matrix D.

6)

0 1 1 -2

A) P = 1 + 2

1-

1

2 0

2 , D = -1 1

B) P = 1 - 2 1 + 2 , D = -1 - 2 1

1

0

C) P = 1 - 2 1 + 2 , D = -1 - 2 1

-1

0

D) P = 1 - 2 1 + 2 , D = 1 - 2 1

1

0

-1 +

0 2

-1 +

0 2

-1 +

0 2

-1 -

0 2

7) Orthogonally diagonalize the matrix, giving an orthogonal matrix P and a diagonal matrix D.

7)

12 -6 -6 7

A) P =

3/ 13 -2/ 13

2/ 13 , D = 16 0 0 3 3/ 13

B) P =

3 -2 , D = 16 0 0 3 -2 3

C) P =

3/ 13 -2/ 13 , D = 3 0 0 16 -2/ 13 3/ 13

D) P =

3 2 , D = 16 0 0 3 -2 3

8) Orthogonally diagonalize the matrix, giving an orthogonal matrix P and a diagonal matrix D. 1 2 2 1

A) P =

C) P =

1 2 -

-

3 0 1 ,D= 0 -1 2

1 2 1 2

1 2

1 2

-

1 2

1 ,D= 2

B) P =

-1 0 0 -1

D) P =

2

1 2

1 2

-

1 2

1 0 1 ,D= 0 -1 2

-

1 2

1 2

1 2

1 0 1 ,D= 0 -1 2

8)

9) Orthogonally diagonalize the matrix, giving an orthogonal matrix P and a diagonal matrix D. 1 2

2 0

1 3

A) P =

2 6

1 ,D= 3

1 3

2 6

-

C) P =

2 6

-1 0 0 -2

B) P =

1 0 1 ,D= 0 2 3

2 6

D) P =

1 3

2 6

1 ,D= 3

2 6

-

1 3

2 6

1 ,D= 3

2 6

-1 0 0 2

-1 0 0 2

10) Orthogonally diagonalize the matrix, giving an orthogonal matrix P and a diagonal matrix D. 13 10 10 12 4 6

4 6 5

2/3 -2/3

1/3

1 0

0

1/3

2/3

0 0 25

A) P = 2/3 1/3 -2/3 , D = 0 4 0 2/3

2 -2 1

1 0 0

1 2 2

0 0 25

2 -2 1

25 0 0

1 2 2

0 0 1

2/3 -2/3

1/3

25 0 0

1/3

2/3

0 0 1

D) P = 2/3 1/3 -2/3 , D = 0 4 0 2/3

11) Orthogonally diagonalize the matrix, giving an orthogonal matrix P and a diagonal matrix D. 11 7 7 7 11 7 7 7 11

1/ 3

1/ 2

-1/ 3

0

1/ 6 25 0 0 , D = 0 25 0 1/ 6 0 0 4 2/ 6

1 -1 -1

25 0 0

1 0 2

0 0 4

B) P = 1 1 -1 , D = 0 4 0 1/ 3

C) P = 1/ 3 1/ 1/ D) P = 1/ 1/

3 3 3 3

-1/ 2 1/ 2 0 -1/ 2 1/ 2 0

-1/ -1/ 2/ -1/ -1/ 2/

10)

B) P = 2 1 -2 , D = 0 4 0

C) P = 2 1 -2 , D = 0 4 0

A) P = 1/ 3 -1/ 2

9)

6 25 0 0 6 ,D= 0 4 0 0 0 4 6 6 4 0 0 6 ,D= 0 4 0 0 0 25 6

3

11)

12) Orthogonally diagonalize the matrix, giving an orthogonal matrix P and a diagonal matrix D. 5 -4 -2 -4 5 -2 -2 -2 8

2 3

-1 3 2

-1 2

A) P = 3

2

-1 3 2

1 3

2 2 3

-1 000 2 ,D= 0 9 0 009 -1

2 3

-1 3 2

1 2

2

-1

1 3

2 2 3

1 000 2 ,D= 0 9 0 009 0

2 3

-1 3 2

-1 2

B) P = 3

2

-1 3 2

1 3

2 2 3

-1 000 2 ,D= 0 9 0 009 0

2 3

-1 3 2

-1 2

2

-1

1 3

2 2 3

1 000 2 ,D= 0 9 0 009 0

2

C) P = 3 3 2

D) P = 3 3 2

13) Orthogonally diagonalize the matrix, giving an orthogonal matrix P and a diagonal matrix D. 0 1 1 1 0 1 1 1 0

A) P = -

C) P =

1 3

-1 6

1 3

-1 6

1 3

2 3

-

1 2 1 2 0 0 2 , D = 0 -1 0 0 0 -1 0

1 3

-1 6

1 2

1 3

-1 6

1 3

2 3

1 2 0 0 2 , D = 0 -1 0 0 0 -1 0

B) P =

D) P =

4

1 3

-1 6

1 2

1 3

1 6

1 3

2 3

1 2 0 0 2 , D = 0 -1 0 0 0 -1 0

1 3

-1 6

1 3

-1 6

1 3

2 3

-

1 2 1 2 0 0 2 , D = 0 -1 0 0 0 -1 0

12)

13)

14) Orthogonally diagonalize the matrix, giving an orthogonal matrix P and a diagonal matrix D. 4 2 2

2 4 2

2 2 4

A) P =

C) P =

1 3

-1 6

1 3

1 6

1 3

2 3

1 3

-1 6

1 3

1 6

1 3

2 3

-

1 2 1 8 0 0 2 ,D= 0 8 0 0 0 2 0

-

B) P =

1 2 1 2 0 0 2 ,D= 0 2 0 0 0 8 0

D) P =

1 3

-1 6

1 3

-1 6

1 3

2 3

1 3

-1 6

1 3

1 6

1 3

2 3

-

14)

1 2 1 8 0 0 2 ,D= 0 2 0 0 0 2 0

-

1 2 1 2 0 0 2 ,D= 0 8 0 0 0 2 0

T

15) Compute the quadratic form x A x for the given matrix A and vector x .

15)

x1 A = -3 0 , x = x2 0 -6

A) -3x 1 - 6x2

B) -6x 1 2 - 3x 2 2

C) -6x 1 - 12x 2

D) -3x 12 - 6x 2 2

T

16) Compute the quadratic form x A x for the given matrix A and vector x .

A=

3 5/2

5/2 , x = x 1 x2 3

A) 3x 1 2 + 3x 22

B) 3x 1 + 3x 2

C) 3x 1 2 + 5x1 x 2 + 2x2 2

D) 5x 1 2 + 3x 22 T

17) Compute the quadratic form x A x for the given matrix A and vector x .

9 A= 3 0

16)

3 0 x 1 5 ,x = y 5 -2 z

A) 9x2 + y2 - 2z2 C) 9x2 - y2 + 2z2 - 6xy - 10yz

B) 9x2 + y2 - 2z2 + 3x + 8y + 5z D) 9x2 + y2 - 2z2 + 6xy + 10yz

5

17)

T

18) Compute the quadratic form x A x for the given matrix A and vector x .

3 A= 1 1

18)

x1 1 1 0 2 , x = x2 2 0 x3

A) 3x 1 2 + x 22 - 2x 3 2 + x1 x 2 + 4x2 x 3 + 2 x 1 x 3 B) 3x 1 2 + x 1x 2 + 4x 2x 3 - 2x 1x 3 C) 3x 1 2 + x 1x 2 + 4x 2x 3 + 2x 1x 3

D) 3x 1 2 + x 22 - 2x 3 2 + 6x 1 x 2 + 10x 2x 3 19) Find the matrix of the quadratic form. 3x 1 2 + 5x 2 2 + 2x3 2 - 12x 1 x 2 + x2 x 3 3 -12 0 5 1 0 1 2

B) -6

3 -6 0 -6 5 1/2 0 1/2 2

D) 0 5 0

3

-6 1/2 5 0 1/2 0 2

A) -12

C)

19)

3

0

0

0

0

2

20) Find the matrix of the quadratic form. 8x 1 x2 - 16x1 x 3 + 6x2 x 3 A)

0 4 -8

4 -8 0 3 3 0

B)

1 4 -8

4 -8 1 3 3 1

4

0

0

0

C) 0 -8

20) 0 0 3

D)

0 8 -16

8 -16 0 6 6 0

21) Find the matrix of the quadratic form. x 1 2 + 4x1 x 2 + 2x2 2 A) 1 2

C) 1 2

B) 0 2

2 1

2 2

2 2

22) Find the matrix of the quadratic form. -2x 1x 2 A)

0 -1

-1 0

B)

1 -1

21) D)

1 -2 -2 2

D)

0 -1

22)

-1 0

C)

6

0 1

1 0

-1 1

23) Find the matrix of the quadratic form. 3x 1 2 + 5x 2 2 + 2x3 2 3 -12 5 0 1

A) -12

0 1 2

23) 3

-6 1/2 5 0 1/2 0 2

B) -6

3 -6 0 5 1/2 0 1/2 2

3

0

0

0

0

2

D) 0 5 0

C) -6

24) Find the matrix of the quadratic form. x 1 2 + x 22 + x 3 2 + 2x1 x 2 + 2x2 x 3 1 0 0

A) 0 1 0 0 0 1

1 1 0

1 0 1

B) 1 1 1

C) 0 1 1

0 1 1

1 1 1

24) 1 1 1

D) 1 1 0 1 0 1

25) Classify the given quadratic form as semidefinite, indefinite, negative definite, or positive definite.

25)

Q( x ) = 4x 1 2 + 7x 2 2 + 4x 1 x 2

A) semidefinite C) negative definite

B) indefinite D) positive definite

26) Classify the given quadratic form as semidefinite, indefinite, negative definite, or positive definite.

26)

Q( x ) = x 1 2 + x 2 2 + 4x 1 x 2

A) semidefinite C) negative definite

B) indefinite D) positive definite

27) Classify the given quadratic form as semidefinite, indefinite, negative definite, or positive definite.

27)

Q( x ) = 13x 1 2 + 12x 2 2 + 5x 2 2 + 5x 3 2 + 20x 1 x 2 + 8x 1 x 3 + 12x 2 x 3

A) semidefinite C) negative definite

B) indefinite D) positive definite

28) Classify the given quadratic form as semidefinite, indefinite, negative definite, or positive definite.

28)

Q( x ) = -4x 1 2 - 5x 2 2 -4x 3 2 + 2x 1 x 2 - 2x 2 x 3

A) semidefinite C) negative definite

B) indefinite D) positive definite

29) Classify the given quadratic form as semidefinite, indefinite, negative definite, or positive definite.

Q( x ) = 4x 1 2 + x 2 2 + x 3 2

A) semidefinite C) negative definite

B) indefinite D) positive definite

7

29)

30) Classify the given quadratic form as semidefinite, indefinite, negative definite, or positive definite.

30)

Q( x ) = -4x 1 2 - x 2 2 - x 3 2

A) semidefinite C) negative definite

B) indefinite D) positive definite

31) Classify the given quadratic form as semidefinite, indefinite, negative definite, or positive definite.

31)

Q( x ) = x 1 2 - x 2 2 - x 3 2

A) semidefinite C) negative definite

B) indefinite D) positive definite

32) Classify the given quadratic form as semidefinite, indefinite, negative definite, or positive definite.

32)

Q( x ) = 4x 1 2 + 3x 2 2 + 8x 1 x 2

A) semidefinite C) negative definite

B) indefinite D) positive definite

33) Classify the given symmetric matrix as indefinite, semidefinite, positive definite, or negative

33)

definite.

2 2 2 5

A) indefinite C) positive definite

B) semidefinite D) negative definite

34) Classify the given symmetric matrix as indefinite, semidefinite, positive definite, or negative

34)

definite.

0 1 1 -2

A) indefinite C) positive definite

B) semidefinite D) negative definite

35) Classify the given symmetric matrix as indefinite, semidefinite, positive definite, or negative definite.

12 -6 -6 7 A) indefinite

B) semidefinite D) negative definite

C) positive definite

8

35)

36) Classify the given symmetric matrix as indefinite, semidefinite, positive definite, or negative

36)

definite.

1 2 2 1

A) indefinite C) positive definite

B) semidefinite D) negative definite

37) Classify the given symmetric matrix as indefinite, semidefinite, positive definite, or negative

37)

definite.

13 10 4 10 12 6 4 6 5 A) indefinite

B) semidefinite D) negative definite

C) positive definite

38) Classify the given symmetric matrix as indefinite, semidefinite, positive definite, or negative

38)

definite.

11 7 7 7 11 7 7 7 11 A) indefinite

B) semidefinite D) negative definite

C) positive definite

39) Classify the given symmetric matrix as indefinite, semidefinite, positive definite, or negative

39)

definite.

0 1 1 1 0 1 1 1 0 A) indefinite

B) semidefinite D) negative definite

C) positive definite

40) Classify the given symmetric matrix as indefinite, semidefinite, positive definite, or negative definite.

0 0 0 0 1 0 0 0 2 A) positive semidefinite

B) indefinite D) negative definite

C) positive definite

9

40)

41) Classify the given symmetric matrix as indefinite, semidefinite, positive definite, or negative

41)

definite.

0 0 0 0 -1 0 0 0 -2 A) negative semidefinite

B) negative definite D) positive definite

C) indefinite

42) Classify the given symmetric matrix as indefinite, semidefinite, positive definite, or negative definite.

5 -4 -2 -4 5 -2 -2 -2 8 A) positive semidefinite

B) positive definite D) indefinite

C) negative definite

10

42)

43) Sketch the graph of the equation on R2. Show both the original axes and the new axes.

5x 2 + 2y 2 +

3 xy = 6 5

A)

B)

C)

D)

11

43)

44) Sketch the graph of the equation on R2. Show both the original axes and the new axes.

44)

5x 1 2 - 3x 2 2 + 6x 1 x 2 = 15

A)

B)

C)

D)

45) Identify the shape of the graph of the quadratic form on R2 .

45)

2x 1 2 + 2x 2 2 - 3x 1 x 2 = 6

A) Circle C) Hyperbola

B) Set of two parallel lines D) Ellipse

46) Identify the shape of the graph of the quadratic form on R2 .

4x 1 2 + 9x 2 2 = 36

A) Hyperbola C) Ellipse

B) Circle D) Set of two parallel lines

12

46)

47) Identify the shape of the graph of the quadratic form on R2 .

47)

4x 1 2 + x 2 2 + 4 x 1 x 2 = 1

A) Ellipse C) Set of two parallel lines

B) Hyperbola D) Circle

48) Identify the shape of the graph of the quadratic form on R2 .

48)

5x 1 2 - 3x 2 2 + 9x 1 x 2 = -1

A) Set of two parallel lines C) Ellipse

B) Circle D) Hyperbola

49) Identify the shape of the graph of the quadratic form on R3 .

49)

x 1 2 - x 2 2 - 4x 2 x 3 - 4x 1 x 3 = -1

A) Ellipsoid C) Hyperbolic cylinder

B) Hyperboloid of two sheet D) Elliptic cylinder

50) Identify the shape of the graph of the quadratic form on R3 .

50)

x 1 2 + 2x 2 2 = 1

A) Hyperboloid of two sheet C) Elliptic cylinder

B) Ellipsoid D) Hyperbolic cylinder

51) Identify the shape of the graph of the quadratic form on R3 .

51)

2x 1 x 2 + 2x 2 x 3 + 2x 1 x 3 = 1

A) Ellipsoid C) Hyperboloid of two sheet

B) Hyperbolic cylinder D) Elliptic cylinder

52) Find the maximum value of Q(x) subject to the constraint x Tx = 1.

52)

Q(x) = 2x 1 2 + 7x 2 2 + 5x 3 2

A) 2

B) 5

C) 14

D) 7

53) Find the maximum value of Q(x) subject to the constraint x Tx = 1.

53)

Q(x) = 14x 1 2 + 14x 2 2 + 18x 3 2 + 26x 1 x 2 + 18x 1 x 3 + 18x 2 x 3

A) 9

B) 36

C) 1

13

D) 25

54) Find a unit vector at which the quadratic form xTAx is maximized, subject to the constraint xTx =

54)

1.

A= 4 2

2 7

B) 1/ 5

A) 1 2

D) -1/ 5

C) -2/ 5

2/ 5

1/ 5

2/ 5

55) Find a unit vector at which the quadratic form xTAx is maximized, subject to the constraint xTx =

55)

1.

8 -6 4 A = -6 9 -2 4 -2 4

1/3

2/ 5

2/3

A) 2/3

B) -2/3

2/3

2

C) -2/ 5

1/3

D) -2 1

1/ 5

56) Find the maximum value of Q(x) subject to the constraints xTx = 1 and xTu = 0, where u is a unit

56)

eigenvector corresponding to the greatest eigenvalue of the matrix of the quadratic form.

2 2 2 Q(x) = 2 x 1 + 7 x 2 + 4 x 3

A) 4

B) 2

C) 7

D) 0

57) Find the maximum value of Q(x) subject to the constraints xTx = 1 and xTu = 0, where u is a unit

57)

eigenvector corresponding to the greatest eigenvalue of the matrix of the quadratic form.

2 2 2 Q(x) = 2 x 1 + 7 x 2 + 4 x 3

A) 0

B) 7

C) 4

D) 2

58) Find the singular value decomposition of the matrix. -3 0 0

58)

0

A) 1 0

3 0

0 0

1 0

0 1

B) 1

0 -1

0 0

0 3

1 0

0 1

C) -1 0

3 0

0 0

1 0

0 1

D) -1 0

0 0

0 3

1 0

0 1

0

0

1

1

0

0

14

1

1 0 -1

59) Find the non-zero singular values of the matrix.

A) 3, 2

B) 1, -1

A)

C)

1 2

1 6

1 3

0

-2 6

1 3

-1 2

1 6

1 3

1 2

1 6

1 3

0

-2 6

1 3

-1 2

1 6

3 0

3 0 0

0 2

0 2 0

59) D) 3, 2

C) -2, -3

60) Find a singular value decomposition of the matrix.

1 3

1 1 1

0 1

1 0

1 0

1 0 -1

1 1 1

B)

0 1

D)

60)

1 3

1 2

1 6

1 3

0

-2 6

1 3

-1 2

1 6

1 3

1 2

1

1 3

0

-2

1 3

-1 2

1

3 0 0

0 2 0

0 1

1 0

3 0 0

0 2 0

0 1

1 0

61) Find the non-zero singular values of the matrix. 4 11 14 8

A) 60, 30

B) 6 10, 3 10

15

61)

7

-2 C) 360, 90

D) 30, 15

62) Find a singular value decomposition of the matrix. 4 11 14

A)

B)

C)

D)

3 10

1 10

1 10

-3 10

-3 10

1 10

1 10

3 10

3 10

1 10

1 10

-3 10

3 10

1 10

1 10

-3 10

8

6 10 0

0 3 10

6 10 0

0

0 0

3 10

6 10 0

0

0 0

3 10

6 10 0

1 2 -2 -1 2 -2

0 0

0 3 10

7

62)

-2

2 2 1

1 3

2 3

2 3

-2 3

-1 3

2 3

2 3

-2 3

1 3

1 3

2 3

2 3

-2 3

-1 3

2 3

2 3

-2 3

1 3

1 3

2 3

2 3

-2 3

-1 3

2 3

2 3

-2 3

1 3

63) Find the pseudoinverse of the matrix [1 2].

A) 1 5

2 5

63)

B) 1 -2 2

C)

1

16

1 5 0

D)

1 5 2 5

Answer Key Testname: UNTITLED8

1) C 2) D 3) B 4) B 5) D 6) B 7) A 8) A 9) B 10) D 11) C 12) D 13) D 14) B 15) D 16) C 17) D 18) C 19) C 20) A 21) C 22) A 23) D 24) B 25) D 26) B 27) D 28) C 29) D 30) C 31) B 32) B 33) C 34) A 35) C 36) A 37) C 38) C 39) A 40) A 41) A 42) A

17

Answer Key Testname: UNTITLED8

43) B 44) B 45) D 46) C 47) C 48) D 49) C 50) C 51) C 52) D 53) B 54) B 55) B 56) A 57) C 58) C 59) A 60) B 61) B 62) C 63) D

18

Exam Name___________________________________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

1) Determine the complex conjugate of 7i - 3.

A) 3 - 7i B) -3 + 7i

1) C) -3 - 7i

2) Determine the complex conjugate of 7i + 5.

A) -5 - 7i B) 5 + 7i

D) 3 + 7i 2)

C) 5 - 7i

D) -5 + 7i

3) Determine the real and imaginary parts of the complex number by first writing the number in standard form. z = (5 - 3i) (5 + 3i) A) Re(z) = 34 and Im(z) = 0

3)

B) Re(z) = 34 and Im(z) = 6 D) Re(z) = 30 and Im(z) = 4

C) Re(z) = 32 and Im(z) = 2

4) Determine the real and imaginary parts of the complex number by first writing the number in standard form. z = (3 - 2i) (-2 + 5i) A) Re(z) = -16 and Im(z) = 11

4)

B) Re(z) = 16 and Im(z) = 19 D) Re(z) = 4 and Im(z) = 11

C) Re(z) = 4 and Im(z) = 19

5) Determine the real and imaginary parts of the complex number by first writing the number in

5)

1 standard form. i

A) Re(z) = -1 and Im(z) = 0 C) Re(z) = 0 and Im(z) = 1

B) Re(z) = 0 and Im(z) = -1 D) Re(z) = 1 and Im(z) = 0

6) Write the number in standard form: (3 - 4i) - (2 - 5i).

A) 1 + i B) 5 + 9i C) 1 - 9i

D) 5 + i

7) Calculate i6 in standard form.

A) -i B) -1

C) 1

D) i

8) Calculate (2 + i)(1 - i) in standard form.

A) 5 - 2i B) 2

C) 3

D) 5

6)

7)

8)

9) Calculate (6 + 2i)i in standard form.

A) 2 + 6i B) -2 - 6i

9) C) -2 + 6i

10) Calculate (3 - 4i) (2 - 3i) - (2 - 3i).

A) 0 B) 3 - 4i

D) 2 - 6i 10)

C) -16 - 14i

1

D) -8 - 14i

11) Calculate (7 - 2i) (-3 + 3i).

A) -15 - 27i B) -21 - 6i

11) C) -27 + 27i

D) -15 + 27i

12) Express 1+ 3i in the standard form.

12)

3+i

A) 3 + 4 i 5

C) 3 - 4 i

B) 1

5

5

D) 1 + i

5

3 in the standard form.

2 - 7i

13) Express

A) 6 + 21 i 53

13)

B) 3 - 3 i

53

2

C) 3 + 3 i

7

2

D) 6 - 21 i

7

53

53

14) Use polar form to calculate: (1 + i)3 and then write your answer in standard form.

A) -2 + 2i B) -2 - 2i C) 2 + 2i D) 2 - 2i

14)

15) Use polar form to calculate (1 + i) (1 + 3 i).

15)

A) 2 cos C) 2 2 cos

12

+ i sin + i sin

12

B) 2 2 cos

12

D) 2 cos

12

12 12

+ i sin + i sin

12 12

16) Use polar form to calculate 1 + i .

1+

A) 2 cos 2

C) 2 cos 2

12

12

+ i sin

+ i sin

16)

3i

B) 2 cos

12

2

D) 2 cos

12

2

12 12

+ i sin + i sin

12 12

17) Use polar form to calculate 1 - 3i 1 + i .

A) 2 2 cos

12

C) 2 2 cos -

12

+ i sin

17) B) 2 2 cos -

12

+ i sin -

D) 2 2 cos

12

18) Use polar form to calculate -1 + 3i 9 .

A) 64 B) -512

12

12

+ i sin + i sin

12

12

18) C) -64

19) Use polar form to calculate 1 + 3i 8 .

A) -128 1 + 3i B) 128 -1 + 3i

D) 512 19)

C) 128 1 - 3i

20) Use polar form to calculate (1 + i)4 .

A) -4i B) -4

D) 128 1 + 3i 20)

C) 4

2

D) 4i

21) Use polar form to calculate (3 - 3i)3.

A) 54 + 54i B) -54 + 54i

21) C) -54 - 54i

D) 54 - 54i

22) Use polar form to find the roots of a complex number (-1)1/6. + 2k 6

A) 2 cos

+ 2k 6

B) cos C) cos

+ 2k 5

D) cos

+ i sin

+ 2k 6

+ i sin

+ i sin

3

+ 2k 6

22)

,0 k 5 ,0 k 5

,0 k 5

3

+ 2k 5

+ i sin

,0 k 5

23) Use polar form to find the roots of a complex number (-i)1/4.

A) cos (4k-1) 8

(4k-1) + i sin 8

,0 k 4

B) cos (4k-1)

+ i sin

(4k-1) 8

,0 k 3

C) cos (2k-1)

+ i sin

(2k-1) 8

,0 k 3

D) cos (2k-1)

+ i sin

(2k-1) 4

,0 k 3

8 8 4

23)

24) Use polar form to find the roots of a complex number (1)1/2.

A) ± 1, i B) ± i C) ±1, ±i

24) D) ± 1

25) Use polar form to find the roots of a complex number (-1)1/3.

A) 1,

1+

2

3 i

, and

1-

2

3 i

,

B) 1,

1+

25) 2i

2

, and

1-

2

2i

D) 1, -1 + 2 i , and -1 - 2 i

C) 1, -1, and i

2

2

26) Use polar form to find the roots of a complex number i 1/2.

26)

A) 1 - i and - 1 - i 2 2

B) 1 + i and - 1 + i 2 2

C) 1 - i and - 1 + i

D) 1 + i and - 1 - i

2

2

2

2

-i

27) Convert z = 3e 3 to standard form.

A) 1 2

3 i 2

27)

B) - 3 - 3 3 i 2

C) + 3 + 3 3 i

2

2

3

2

D) 3 - 3 3 i 2

2

i

28) Convert z = 4e 3 to standard form.

B) - 1 +

A) -2 - 2 3i

2

28) 3 i 2

C) 1 + 2

3 i 2

D) -2 + 2 3i

29) Find the general solution to the system: ix + y = 3 ix - y = 2

A) x = y

-

5 i 2

-

1 2

-

B) x = y

29)

5 i 2

-

C) x =

1 2

y

5 i 2

5 i 2

D) x = 1 y

1 i 2

2

30) Find the general solution to the system:

30)

-ix + y = 0 x + iy = 0

A) x = - i t, t

C

B) x = - i t, t

C

C) x = - i t, t

C

D) x = - i t, t

C

y

y

1

-1

y

y

1+i

1

31) Find the general solution to the system:

31)

-ix + y = 2 x + iy = 1

A) No solution C) x = - i t, t y

1

B) x = - i t, t

C

D) x = - i t, t

C

y

C

y

4

1+i

1

32) Find the general solution to the system:

32)

(1 - i) x + y = 0 2x +(1 + i) y = 0

A) x = 1 + i t, t C

B) x = -1 - i t, t C

C) x = 1 - i t, t C

D) x = 1 - i t, t C

y

y

y

2

2

y

-2

1

33) Find the general solution to the system:

33)

(1 + i) z1 + 2i z2 = 1 (1 + i) z2 + z3 =

1 1 - i 2 2

z 1 - z3 = 0

0 1 1 (-1 - i) +t z2 = 2 2 z3 0 1 z1

A)

B)

0 1 1 (-1 - i) +t z2 = 2 2 z3 0 1 z1

C)

0 1 i (-1 + i) +t z2 = 2 2 z3 0 1 z1

0 1 1 (-1 + i) +t z2 = 2 2 z3 0 1 z1

D)

34) Calculate the following:

34)

3+i i 2 - i - -i -1 1-i

A)

3 2 2-i

3

3

B) 2

C) 2 - 2i

1

1

5

3

D) 2 - 2i 2-i

35) Calculate the following:

35)

2 + 2i 1-i 2-i + 3i -1 1-i

3+i

A) 2 + 2i

3+i

3-i

B) 2 + 2i

C) 2 - 2i

i

-i

3+i

D) 2 + 2i i

-i

36) Calculate the following: -3i

36)

i 1+i 1 - 2i

A)

3 3 - 3i -6 - 3i

3

-3

B) -3 - 3i

C) 3 - 3i

-6 - 3i

D)

6 - 3i

-3 3 - 3i -6 - 3i

37) Calculate the following:

37)

-2i 1 + 2i 2 - 3i

A) 4 - 2i 6 - 4i

B) -4 - 2i

C)

-6 - 4i

1

i

-1

i

4 - 2i -6 - 4i

D) 4 + 2i 6 + 4i

38) Find a basis for the column space of A = 1 + i -1 + i .

A)

C)

1 i 1 + i , -1 + i i -1

1

i

-1

i

38)

B)

i -1 + i i

D)

1 1+i -1

39) Find a basis for the row space of A = 1 + i -1 + i .

A)

B)

0 1

39) C)

6

1 0

D)

1 , 0 0 1

1

i

-1

i

40) Find a basis for the nullspace of A = 1 + i -1 + i .

40)

A)

0 0

B)

1 1

C)

1 i 1 + i , -1 + i i -1

D)

1 , 0 0 1

0

41) Find a basis for the column space of A = 1 + i A)

0 1 1 1+i , 1 , i i 1+i -i

C)

1 1 1 , i i 1+i

-i

0

1 1 i

42) Find a basis for the row space of A = 1 + i A)

1 0 0 0 , 1 , 0 0 0 1

B)

1 0 0 , 1 1 1

-i

1 1 i

1 i .

1+i

41)

B)

0 1 1+i , i -i 1 + i

D)

0 1 1+i , 1 i -i

1 i .

1+i

42)

C)

0 1

43) Find a basis for the nullspace of A = 1 + i 1 -i

i

1 i .

1+i

D)

43)

A)

1 , 0 0 1

B)

1 i , 1 + i -1 + i i -1

C)

1 1

D)

i -1 1

i i 1 , v = i . Then find u , v .

-1 + i -1 A) 0 B) 2

1 0 1

44) Let u =

44) C) 2 - 2i

7

D) 2i

i 1 . Then find u .

-1 + i A) 1 B) 2

45) Let u =

46) Let v =

45) C) 4

D) 0

i i . Then find v .

-1

A) 1

46) C) 3

B) 2

D) 2

1 0 i 47) Let v = , and u = 1 . Then find proj u v .

1 -i 0 i 0 i 1 A) 3 -i i

47)

0 4i 1 B) 3 -i i

2

i

3 orthogonal in C3 .

k

0 2 1 C) 3 -i i

0 2i 1 D) 3 -i i

48) Let v = 1 , and u = 4i . Then find all complex values of k (if any) such that v and u are

B) 2

A) -2i

C) 4i

3

3

D) 2i

49) Which of the following matrices are unitary?

A=

1 1-i 3 1

48)

49)

-i and B = 1 i -1 + i -i 1

A) Neither A nor B C) A only

B) B only D) Both A and B

50) Which of the following statements are true for a unitary matrix A? I) Det A = ±1.

II) All of the eigenvalues satisfy = 1.

A) Neither I nor II B) Both I and II

C) II only

8

50) D) I only

51) Use the Gram-Schmidt procedure to find an orthogonal basis for S. S = Span

0 1 1 , i -i 1 i 0

.

0 3 1 , i -i 1 i 2

A)

0 3 1 , -i 1 -i i 2

B)

1 + 2i

52) Determine the projection of z = 2 + i 1 i , 0 1 i 1

vectors

-

0 3 1 , i i -i i 2

C)

0 3 1 , i 1 -i i 2i

D)

onto the subspace C3 spanned by the orthogonal set of

.

1 13 + i 3 6

-

1 7 + i 3 6

-

5 1 + i 3 3

B)

7 1 - i 6 3

C)

13 1 - i 6 3

1 7 + i 3 6

1 7 - i 3 6

5 1 + i 3 3 13 1 + i 6 3

D)

5 1 + i 3 3 13 1 + i 6 3

1

53) Determine the projection of z = i onto the subspace C3 spanned by the orthogonal set of 1 i 0 , 1 i 1

vectors

0

.

1 3

1 2

A) 0

B) 0

i 2

52)

1-i

5 1 - i 6 6

A)

51)

1

C) 0

i 2

i

9

0

D) 0 0

53)

54) Find the eigenvalues of A, and find a basis for each eigenspace.

54)

A = 0.76 -1.04 0.64 0.44

1 + 5i 4 1 + 5i 4

; -0.6 + 0.8i,

C) 0.6 + 0.8i,

1 - 5i 4

; 0.6 - 0.8i,

1 + 5i 4

D) 0.6 - 0.8i,

1 - 5i 4

; 0.6 + 0.8i,

1 + 5i 4

A) -0.6 - 0.8i, B) -0.6 + 0.8i,

; -0.6 - 0.8i,

1 - 5i 4 1 - 5i 4

55) Find the eigenvalues of A, and find a basis for each eigenspace.

A=

55)

2 4 -4 2

A) 2 - 4i,

1 -i

; 2 + 4i,

1 i

B) 2 - 4i,

1 - 2i -4

; 2 + 4i,

1 + 2i -4

C) 2 - 4i,

1 i

; 2 + 4i,

1 -i

D) 2 - 4i,

1 + 2i -4

; 2 + 4i,

1 - 2i -4

56) Find a real canonical form of a matrix C of A = -1 2 and find a change of coordinates matrix P -1 -3

such that P-1 AP = C. A) C = -2 1 and P = -1 -1 1 0 -1 2

B) C = -2 1 and P = 0 -1

C) C = -2 1 and P = 0 -1

D) C = -2 1 and P = -1 -1

-1 -2

-1 2

1 0

-1 -2

10

1 0

1 0

56)

57) Find a real canonical form of a matrix C of A = -2 -1 and find a change of coordinates matrix P 5 2

such that P-1 AP = C. 2 1 A) P = 5 5 , and C = 1

C) P =

-

0

2 5

1 5 , and C =

1

0

2

0 1 -1 0

B) P = 5

0 1 -1 0

D) P =

58) Find a real canonical form of a matrix C of A =

3 0 0

1 1 0

0 -2 1

1 5 , and C =

1

0

2 5

-

1

0 1 -1 0

1 5 , and C =

0 1 -1 0

0

-1 -4 -4 4 7 4 and find a change of coordinates matrix 0 -2 -1

P such that P-1 AP = C. 1 1 1 3 0 0 A) P = -2 -1 1 , and C = 0 1 2 1 1 0 0 -2 1

1 1 1

-

-

1 1 1

3 0 0

1 1 0

0 -2 1

1 1 1

3 0 0

1 1 0

0 -2 1

D) P = -2 -1 -1 , and C = 0 1 2

59) Which of the following matrices are Hermitian?

1-i ,B= 1 -i -i

59)

i 2+i

A) Both A and B C) A only

B) B only D) Neither A nor B 1 2 + ki .

2+i 2 D) k = -1

60) Find value(s) of k (k should be a real number) such that A is Hermitian., where A = A) k = 2

58)

B) P = 2 -1 -1 , and C = 0 1 2

C) P = -2 1 -1 , and C = 0 1 2

A= 2 1+i

57)

B) k = 1

C) k = 0

11

60)

61) Find a unitary matrix U and a diagonal matrix D such that U AU = D. A=

2 -i

i 2 i

-

1+i 2

i 2

A) U =

1 2 i

C) U =

61)

2 1 2

2

1 1 ,D= 0 2 1+i 2

0 3

B) U = 1 - i

0 1

D) U =

,D= 1 0

0 3

1 1-i ,D= 0 2

0 3

1

2

2

2

1+i 2

i

3 1-i ,D= 0 2

i 2

1 2

62) Find a unitary matrix U and a diagonal matrix D such that U AU = D. A=

4 1+i

A) U =

C) U =

62)

1-i 5 1+i 6 2 6 1-i 6 2 6

-1 + i 3

6 1 ,D= 0 3

-1 + i 3

6 1 ,D= 0 3

0 3

B) U =

0 3

D) U =

12

1-i 6 2 6 1-i 6 -

2 6

-1 + i 3

3 1 ,D= 0 3

-1 + i 3

6 1 ,D= 0 3

0 6

0 3

63) Find a unitary matrix U and a diagonal matrix D such that U AU = D. 5 A= 0 0

0 -1 -1 - i

A) U =

D=

C) U =

D=

0 -1 + i 0

0 2 6

0 -1 + i 6

1+i 6

2 6

-2 0 0

0 1 0

0 -1 + i 6

1+i 6

2 6 0 -1 0

1

0 2 6

0 -1 + i 6

1+i 6

2 6

0 ,

B) U =

0

0 0 5

0 2 6

-2 0 0

63)

2 D= 0 0 1 0 ,

D) U =

0

0 0 5

D=

13

0 1 0

0 -1 + i 6

1+i 6

2 6 0 1 0

0 , 0

0 0 5

0 2 6

-2 0 0

1

0 0 -5

1 0 , 0

Answer Key Testname: UNTITLED9

1) C 2) C 3) A 4) C 5) B 6) A 7) B 8) D 9) C 10) D 11) D 12) A 13) A 14) A 15) C 16) A 17) B 18) D 19) B 20) B 21) C 22) B 23) B 24) D 25) A 26) B 27) D 28) D 29) B 30) D 31) A 32) B 33) B 34) A 35) A 36) A 37) C 38) C 39) D 40) A 41) D 42) B

14

Answer Key Testname: UNTITLED9

43) D 44) C 45) B 46) C 47) D 48) A 49) C 50) C 51) A 52) A 53) B 54) D 55) A 56) D 57) C 58) D 59) D 60) D 61) A 62) C 63) A

15