Elementary Linear Algebra Applications Version, Enhanced eText, 12th Edition, Howard Anton , Chris R

Type:

Test Bank

Elementary Linear Algebra Resource: Applications Version, Enhanced eText Edition:

12th Edition

Author(s):

Howard Anton Chris Rorres Anton Kaul

Chapter 1: Systems of Linear Equations and Matrices Multiple Choice Questions 1. Which of the following equations is linear?

(A) 2x21 + 3x32 + 4x43 = 5 √ √ (B) 3x1 − 2x2 + x3 = 5 √ √ (C) 5x1 + 5 x2 − x3 = 1 (D) 22 x1 + cos (x2 ) + 4x3 = 7 2. Which system corresponds to the following augmented matrix?

1 11 6 3 9 (A)

(B)

(C)

4

0

−2

x1 + 11x2 = −3 9x1 + 4x2 = −2 x1 + 11x2 + 6x3 =

3

= −2

9x1 + 4x2

x1 + 11x2 + 6x3 + 3x4 = 0 − 2x4 = 0

9x1 + 4x2 x1 + 9x2 = 0

(D)

11x1 + 4x2 = 0 6x1

=0

3x1 − 2x2 = 0 3. Which of the following statements best describes the following augmented matrix? ⎡ ⎤ 1 2 6 5 ⎢ ⎥ ⎢ A = ⎣−1 1 −2 3⎥ ⎦ 1 −4 −2 1 (A) A is consistent with a unique solution.

(B) A is consistent with infinitely many solutions.

(C) A is inconsistent.

(D) none of the above.

Elementary Linear Algebra 12e

–2–

Anton/Rorres

4. Which of the following matrices is in reduced row echelon form? ⎡ ⎤ 1 0 −1 1 ⎢ ⎥ ⎥ (A) ⎢ 0 1 2 0 ⎣ ⎦ 0 1 3 1 ⎡ ⎤ 1 0 2 5 ⎢ ⎥ ⎢ (B) ⎣0 1 −7 5⎥ ⎦ 0 0 1 14

1 0 0 11 −3 (C) 0 0 0 1 4 ⎡ ⎤ 1 0 −5 ⎢ ⎥ ⎥ (D) ⎢ 0 1 3 ⎣ ⎦ 0 0 0 5. If the matrix A is 4 × 2, B is 3 × 4, C is 2 × 4, D is 4 × 3, and E is 2 × 5, which of the following expressions is not defined?

(A) AT D + CB T

(B) (B + DT )A

(C) CA + CB T

6. What is the second row of the product AB? ⎤ ⎡ ⎡ 2 0 2 3 ⎥ ⎢ ⎢ ⎥ ⎢ A=⎢ ⎣5 4 8 ⎦ , B = ⎣ 6 2 9 7 2 (A) 18

33

25

(B) 64

48

91

(C) 50

(D) DBAE

1

7

⎤

3

⎥ 2⎥ ⎦

9

7 89

(D) 48

99

7. Which of the following is the determinant of the 2 × 2 matrix A = (A) ad − bc

(B) bc − ad

1 (C) bc−ad

a

b

c

d

1 (D) ad−bc

8. Which of the following matrices is not invertible?

3 6 7 7 9 0 9 (A) (B) (C) (D) 2 4 2 3 4 4 6

3

5

9. Which of the following matrices is not an elementary matrix? ⎡ ⎤ ⎡

0 1 0 1 0 ⎢ ⎥ ⎢ 1 1 1 0 ⎥ ⎢ (C) ⎢ (B) (A) ⎣1 0 0⎦ (D) ⎣0 −2 0 2 5 1 0 0 1 0 0 Chapter 1

0

⎤

⎥ 0⎥ ⎦ 1

89

?

33

Elementary Linear Algebra 12e

–3–

Anton/Rorres

10. For which elementary matrix E will the equation EA = B hold? ⎡ ⎤ ⎡ ⎤ 1 4 6 1 4 6 ⎢ ⎥ ⎢ ⎥ ⎥ , B = ⎢0 0 ⎥ A=⎢ 0 0 1 1 ⎣ ⎦ ⎣ ⎦ 2 10 9 0 2 −3 ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎡ 0 1 0 0 1 0 0 1 0 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ (A) ⎢ ⎣0 1 0⎦ (B) ⎣0 0 1⎦ (C) ⎣ 0 1 0⎦ (D) ⎣0 1 −2 0 1 0 1 0 2 0 1

0

1

⎤

1

⎥ 0⎥ ⎦

0

0

11. Which matrix will be used as the inverted coefficient matrix when solving the following system?

3x1 + x2 = 4

(A)

2

−1

−5

3

(B)

−2 5

5x1 + 2x2 = 7

2 1 1 (C) 5 3 −3

(D)

−2

−1

−5

−3

12. What value of b makes the following system consistent?

4x1 + 2x2 = b 2x1 + x2 = 0 (A) b = −1

(B) b = 0

(C) b = 1

(D) b = 2

13. If A is a 3 × 3 diagonal matrix, which of the following matrices is not a possible value of Ak for some integer k? ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 0 0 0 1 0 1 1 0 0 0 0 0 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ (B) ⎢0 16 ⎥ (C) ⎢0 1 ⎥ (D) ⎢0 0 0⎥ (A) ⎢ 0 4 0 0 0 ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ 4 0 0 9 4 0 25 0 0 −1 0 0 0 ⎤ ⎡ 3 0 0 ⎥ ⎢ ⎥ 14. The matrix ⎢ ⎣0 −7 0⎦ is: 0 0 1 (A) upper triangular.

(B) lower triangular.

(C) both (A) and (B).

(D) neither (A) nor (B).

Chapter 1

Elementary Linear Algebra 12e

–4–

Anton/Rorres

15. If A is a 4×5 matrix, find the domain and codomain of the transformation TA (x) = Ax. (A) Not enough information (B) Domain: R4 , Codomain: R5 (C) Domain: R5 , Codomain: R5 (D) Domain: R5 , Codomain: R4 16. Which of the following is a matrix transformation?

(A) T (x, y, z) = (yx2 , yz 2 ) (B) T (x, y, z, w) = (xy, yz, zw, wx) (C) T (x, y, z) = (x + 1, x + 2, x + z, y + z) (D) T (x, y) = (4x, 5x, −x, 0) 17. Which matrix represents reflection about the xy-plane? ⎤ ⎤ ⎡ ⎤ ⎡ ⎡ 1 0 0 1 0 0 −1 0 0 ⎥ ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎥ (C) ⎢0 1 ⎥ (B) ⎢0 −1 (A) ⎢ 0 0 0 1 0 ⎦ ⎦ ⎣ ⎦ ⎣ ⎣ 0 0 −1 0 0 −1 0 0 1

⎡

−1

0

0

⎤

⎢ (D) ⎢ ⎣ 0

−1

⎥ 0⎥ ⎦

0

0

1

18. Use matrix multiplication to find the image of the vector 2, 1 when it is rotated counterclockwise about the origin through an angle θ = 45◦ . √ √ √ √ √ √ √ √ 2 3 2 3 2 2 2 3 2 (B) 2 , 2 (C) − 2 , 2 (D) − 3 2 2 , 22 (A) 2 , 2 19. Which of the following pairs of operators T1 , T2 : R2 → R2 commute? (That is, for which pair is it true that T1 ◦ T2 = T2 ◦ T1 ?)

(A) T1 is the reflection about the x-axis.

T2 is the reflection about line y = x.

(B) T1 is the orthogonal projection onto the x-axis.

T2 is the reflection about line y = x.

(C) T1 is the counterclockwise rotation about the origin through an angle of π.

T2 is the projection onto the y-axis.

(D) T1 is the reflection about the x-axis.

T2 is the counterclockwise rotation about the origin through an angle of π/2.

Free Response Questions 1. Find the relationship between a and b such that the following system has infinitely many solutions. −x + 2y = a −3x + 6y = b

Chapter 1

Elementary Linear Algebra 12e

–5–

Anton/Rorres

2. Solve the following system and use parametric equations to describe the solution set. x1 + 2x2 + 3x3 = 11 2x1 − x2 + x3 = 2 3x1 + x2 + 4x3 = 13 3. Determine whether the following system has no solution, exactly one solution, or infinitely many solutions.

2x1 + 2x2 = 2 x1 + x 2 = 4

15 −3 4. Find the value of k that makes the system −10 k

6 9

inconsistent.

5. Solve the following system using Gaussian elimination. x1 − x2 − 5x3 = −1 −2x1 + 2x2 + 11x3 = 3x1 − x2 +

x3 =

1 3

6. Solve the following system for x, y, and z.

1 − y1 − z1 = 0 x 2 + y1 + z1 = 3 x 3 − z1 = 0 x

7. The curve y = ax3 + bx2 + x + c passes through the points (0, 0), (1, 1), and (−1, −2).

Find and solve a system of linear equations to determine the values of a, b, and c.

8. Solve the following system for x and y. x2 + y 2 = 6 x2 − y 2 = 2 9. Given C =

1

−1

2

0

, find CC T .

10. Express the following matrix equation as a system of linear equations. ⎤⎡ ⎤ ⎡ ⎤ ⎡ 0 x −1 7 0 ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ 0 4 3⎥ ⎦ ⎣ y ⎦ = ⎣0 ⎦ ⎣ 0 z 6 0 −2

Chapter 1

Elementary Linear Algebra 12e

–6–

Anton/Rorres

11. Find the 3 × 3 matrix A = [aij ] whose entries satisfy the condition aij = i2 − j.

12. Let A and B be n × n matrices. Prove that tr (c · A − B) = c · tr (A) − tr (B).

4 0 13. What is the inverse of ?

9 2 14. Given the polynomial p(x) = x2 − 3x + 1 and the matrix A =

4

4

6

1

, compute p(A).

15. Let A, B, C, and D be n × n invertible matrices. Solve for A given that the following equation holds.

C 2 DA−1 CB −1 = BCB −1 16. Prove that for any m × n matrices A and B, (A − B)T = AT − B T .

17. Use the inversion algorithm to find the inverse of the following matrix. ⎤ ⎡ 1 2 1 ⎥ ⎢ ⎢ 0 2 2⎥ ⎦ ⎣ 0 0 4 18. Which elementary row operation will transform the following matrix into the identity matrix? ⎡ ⎤ 1 0 0 0 ⎢ ⎥ ⎢0 1 0 0⎥ ⎢ ⎥ ⎢ ⎥ ⎢0 ⎥ 0 1 0 ⎣ ⎦ 0 −9 0 1 19. Find the 3 × 3 elementary matrix that adds c times row 3 to row 1.

20. Find the elementary matrix E that satisfies ⎡ ⎤ ⎡ 1 4 6 1 ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ E ⎣0 0 1⎦ = ⎣ 0 2 10 9 0

⎤ 4

6

0

⎥ 1⎥ ⎦

2

−3

21. Solve the following system by inverting the coefficient matrix.

7x + 2y = 1 3x + y = 5

Chapter 1

Elementary Linear Algebra 12e

–7–

Anton/Rorres

22. Solve the following matrix equation for X. ⎤ ⎤ ⎡ ⎡ 2 2 3 0 1 2 3 ⎥ ⎥ ⎢ ⎢ ⎢0 1 4 ⎥ X = ⎢ 0 0 0 1 ⎥ ⎦ ⎦ ⎣ ⎣ 3 1 1 1 5 6 0 ⎡ ⎤ ⎡ ⎤ 1 0 0 3 ⎢ ⎥ ⎢ ⎥ −1 2 ⎥ ⎢ ⎥ 23. Given that A = ⎢ ⎣0 2 1⎦ and b = ⎣1⎦, solve the system A x = b.

0 1 0 2 24. Find a nonzero solution to the following equation.

1 3 x = 3x 4 −3 25. Find the values of a, b, and c that make the following matrix symmetric. ⎡ ⎤ 3 a 2−b ⎢ ⎥ ⎢ 4 0 a + b⎥ ⎣ ⎦ 2 c 7 ⎡ ⎤ ⎡ ⎤ 3 4 3 1 −7 6 ⎢ ⎥ ⎢ ⎥ ⎥, B = ⎢−4 ⎥, and AB = [cij ].

26. Let A = ⎢ 0 0 6 5 0 ⎣ ⎦ ⎣ ⎦ 0 0 2 1 0 2 Find the diagonal entries c11 , c22 , and c33 .

27. Let the entries of a matrix A = [aij ] be defined as aij = 2i2 − i + j + g(j), where g is a function of j. If A is a symmetric matrix, what is g(j)?

28. Prove that for any square matrix A, the matrix B = (A + AT ) is symmetric.

29. Find the domain and codomain of the transformation defined by ⎡ ⎤ x1

⎢ ⎥ ⎢ 5 7 6 0 ⎢ x2 ⎥ ⎥ ⎢ ⎥ ⎥ 1 0 −2 −2 ⎢ ⎣ x3 ⎦ x4 30. Find the standard matrix for the operator T : R2 → R2 defined by 3x1 + x2 = w1 4x2 = w2

Chapter 1

Elementary Linear Algebra 12e

–8–

Anton/Rorres

31. Find the standard matrix for the transformation T defined by the formula T (x1 , x2 , x3 ) = (x1 , −x3 , x2 − x1 , 3x2 + x3 ) 32. Find the standard matrix A for the linear transformation T : R2 → R2 for which

5 −1 3 2 = ,T = T 2 1 −4 −1 33. Prove that the composition of two rotation operators about the origin of R2 is another rotation about the origin.

34. Prove that if TA : R3 → R3 and TA (x) = 0 for every vector x in R3 , then A is the 3 × 3 zero matrix.

35. Write a balanced equation for the following chemical reaction.

C3 H8 + O2 → H2 O + CO2 36. Find the quadratic polynomial whose graph passes through the points 0, 3 , −1, 8 , and 1, 0 .

37. Use matrix inversion to find the production vector x that meets the demand d for the consumption matrix C. ⎤ ⎡ ⎤ ⎡ 18 0.1 0.3 0.2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ C=⎢ ⎣0.5 0.1 0.2⎦ ; d = ⎣40⎦ 26 0.2 0.4 0.3

Chapter 1

Elementary Linear Algebra 12e

–9–

Answers Multiple Choice Answers 1. (B) 2. (B) 3. (C) 4. (D) 5. (C) 6. (C) 7. (A) 8. (A) 9. (B) 10. (C) 11. (A) 12. (B) 13. (B) 14. (C) 15. (D) 16. (D) 17. (C) 18. (A) 19. (C) Free Response Answers 1. 3a = b Chapter 1

Anton/Rorres

Elementary Linear Algebra 12e

– 10 –

2. x1 = −t + 3 , x2 = −t + 4 , x3 = t 3. no solution 4. k = 2 5. x1 = 5, x2 = 11, x3 = −1 6. x = 1, y = − 12 , z = 13

7. System:

c=

0

a+b+c=

0

−a + b + c = −1 Solution: a = 12 , b = − 12 , c = 0 √ 8. x = ±2, y = ± 2

9. CC T =

2

2

2

4

−x + 7y 10.

=0

4y + 3z = 0 − 2z = 0

6x ⎡

−1

0

⎢ 11. A = ⎢ ⎣3 8

13.

14.

1 4 − 98

0

−2

⎤

2

⎥ 1⎥ ⎦

7

6

1 2

29

8

12

23

15. A = B −1 C 2 D ⎡ ⎤ 1 1 −1 4 ⎢ ⎥ 1 1⎥ ⎢ 17. ⎣0 − 2 4⎦ 1 0 0 4 Chapter 1

Anton/Rorres

Elementary Linear Algebra 12e

– 11 –

Anton/Rorres

18. Add 9 times row 2 to row 4 ⎤ ⎡ 1 0 c ⎥ ⎢ ⎥ 19. ⎢ ⎣0 1 0 ⎦ 0 0 1 ⎡

1

0

⎤

⎢ 20. E = ⎢ ⎣ 0

0 1

⎥ 0⎥ ⎦

−2

0

1

21. x = −9, y = 32 ⎡

−33

⎢ 22. ⎢ ⎣ 28

⎤

−43

−67

36

56

⎥ −19⎥ ⎦

−9

−14

5

−7

23

⎡ ⎤ 3 ⎢ ⎥ ⎥ 23. x = ⎢ ⎣9 ⎦ 4

24. Any x =

x1 x2

such that 2x1 = 3x2 . Possible solution: x =

25. a = 4, b = 0, c = 4 26. c11 = −10, c22 = 0, and c33 = 4 27. g(j) = 2j 2 − 2j 29. Domain: R4 , Codomain: R2 30.

3

1

0

4

⎤

⎡ 1

⎢ ⎢ 0 ⎢ 31. ⎢ ⎢−1 ⎣ 0

0 0 1 3

0

⎥ −1⎥ ⎥ ⎥ 0⎥ ⎦ 1

Chapter 1

3 2

Elementary Linear Algebra 12e

8 13 32. −2 0

– 12 –

35. C3 H8 + 5O2 → 4H2 O + 3CO2 36. 3 − 4x + x2 ⎡

⎤ 91.85

⎢ ⎥ ⎥ 37. x ≈ ⎢ 125.50 ⎣ ⎦ 135.10

Chapter 1

Anton/Rorres

Chapter 2: Determinants Multiple Choice Questions 1. What is M23 for the following matrix? ⎡ 3 ⎢ ⎢−1 ⎣ −2 (A) 0

(B) −4

(C) 21

1

0

0

0 (B) −1

0

⎤

2

⎥ 7⎥ ⎦

−4

4

(D) 26

2. Compute.

(A) −40

4

(C) 6

−2

3

5

−6

0

−8

0

0

−4

7

9

1

(D) 0

3. Let A be a square matrix. If det (A) = 5, what is det AT ?

(A) − 15

(B) 15

(C) −5

(D) 5

4. Evaluate the determinant of the following matrix by using signed elementary products. ⎡ ⎤ 0 −1 0 ⎢ ⎥ ⎢2 0 0⎥ ⎣ ⎦ 0 0 8 (A) −16

(B) 16

(C) −9

(D) 9

5. If A is a 3 × 3 matrix and det (A) = 7, what is det (2A)?

(A) 72

(B) 14

(C) 42

(D) 56

6. If A is an invertible matrix and det (A) = 7, what is det (A−1 )?

(A) 7

(B) −7

(C) 17

(D) − 17

Free Response Questions 1. Evaluate.

1

4

0

3 2 1

−2

5

1

Elementary Linear Algebra 12e

–2–

2. Find all values of λ for which

1 − λ

0

0

1 2−λ −1

Anton/Rorres

0

0 = 0

4 − λ

3. Compute the determinant of the following matrix. Simplify completely. ⎤ ⎡ sin (α) cos (β) r · cos (α) cos (β) −r · sin (α) sin (β) ⎥ ⎢ ⎢ sin (α) sin (β) r · cos (α) sin (β) r · sin (α) cos (β)⎥ ⎦ ⎣ cos (α) −r · sin (α) 0 4. If det (A) = −4, find all possible values of k, where A=

1

k

k

3k

.

5. Justify, without evaluating, that the determinant of the following matrix is zero. ⎤ ⎡ 1 0 2 4 ⎢ ⎥ ⎢−2 3 ⎥ 8 6 ⎥ ⎢ ⎥ ⎢ ⎢−1 3 10 10⎥ ⎦ ⎣ 6 6 −3 7 6. Evaluate the determinant of the following matrix by reducing to row-echelon form. ⎤ ⎡ 1 2 8 ⎥ ⎢ ⎥ ⎢2 5 −3 ⎦ ⎣ 8 19 5 7. Let E be a 3 × 3 elementary matrix that interchanges two rows of I3 . Prove that det (E) = −1.

8. If the system Ax = b is inconsistent and A is a square matrix, what is det AT ?

9. Use Cramer’s Rule to solve the following system of equations. x+ y− z =2 3x − y + z = 5 3x + 2y + 4z = 0

Chapter 2

Elementary Linear Algebra 12e

–3–

Anton/Rorres

10. Use the determinant to determine whether the following matrix is invertible. ⎤ ⎡ 7 1 −1 ⎥ ⎢ ⎥ ⎢ 2 4 3 ⎦ ⎣ −14 −2 2 11. Compute the inverse of the following matrix using the adjoint matrix. ⎤ ⎡ 1 2 3 ⎥ ⎢ ⎢2 −1 0⎥ ⎦ ⎣ 3 0 3 12. Prove that if AAT = I, then det (A) = ±1.

13. Let A, B, and P be n × n matrices where at least P is invertible. Prove that if A = P −1 BP , then det (A) = det (B).

Chapter 2

Elementary Linear Algebra 12e

–4–

Anton/Rorres

Answers Multiple Choice Answers 1. (B) 2. (A) 3. (D) 4. (B) 5. (D) 6. (C) Free Response Answers 1. −23 2. λ = 1, 2, 4 3. r2 · sin (α) 4. k = 4 or k = −1 5. The third row is a linear combination of the first two rows.

6. −2 8. 0 9. x = 74 , y = − 17 , z = − 23 24 24 10. The determinant equals 0. Not invertible. ⎡

−3

−6

⎢ 11. A−1 = − 16 ⎢ ⎣−6

−6

3

6

⎤ 3

⎡

1 2

⎥ ⎢ ⎢ 6⎥ ⎦=⎣ 1 −5

− 12

1 1 −1

− 12

⎤

⎥ −1⎥ ⎦ 5 6

Chapter 2

Chapter 3: Euclidean Vector Spaces Multiple Choice Questions

1. Let u = 1, 0, −2 and v = 3, 2, 4 . Compute 2u + v.

(A) 4, 2, 2

(B) 5, 2, 0

(C) 6, 5, 0

(D) 7, 2, 2

2. What

is the terminal point of a vector with initial point A −1, 0 that is equivalent to u = 3, 4 ?

(A) 3, 3

(B) 4, 4

(C) 3, 4

(D) 2, 4 3. Let v = 3. Which of the following vectors is a possible value of v?

(A) 1, 4, 0

(B) −1, −2, 0

(C) 3, 3, 3

(D) 2, −2, 1 4. If u, v, and w are vectors in Rn , which of the following expressions does not make mathematical sense?

(A) (u + v) · w (B) u · (v · w) (C) u · v + w (D) k u − v 5. Which of the following pairs of vectors are orthogonal?

(A) u = 1, 1, −1 , v = 2, 2, 2

(B) u = 1, 0, −1 , v = 2, 0, 1

(C) u = 6, 1, −5 , v = 2, 3, 3

(D) u = 6, −1, −5 , v = 2, 3, 3

Elementary Linear Algebra 12e

–2–

Anton/Rorres

6. The point-normal equation 2(x + 1) + y + (z − 2) = 0 goes through the point P and is normal to n. What are P and n?

(A) P −1, 0, 2 , n = 2, 1, 1

(B) P 2, 1, 1 , n = −1, 0, 2

(C) P −2, −1, −1 , n = −1, 0, 2

(D) P 1, 0, −2 , n = 2, 1, 1 7. Which of the following equations for the line is the correct pair of vector and parametric

containing the point P 4, 0, 2 and parallel to v = 1, −4, 2 ?

(A) Parametric: x = 4 +

t, y = −4t, z = 2 + 2t Vector: x, y, z = 4, 0, 2 + t 1, −4, 2 (B) Parametric: x = 1 + 4t, y = −4,

z = 2 + 2t Vector: x, y, z = t 1, −4, 2 (C) Parametric: x = t, y = −4t, z = 2t

Vector: x, y, z = t 1, −4, 2 (D) Parametric: x = 1, y = −4, z = 2

Vector: x, y, z = 4, 0, 2 + t 1, −4, 2

8. Given the equation of the line x = 8 − t, 6 + 2t , which of the following points is not on the line?

(A) 6, 10 (B) 7, 9 (C) 3, 16 (D) 8, 6 9. If u, v, and w are vectors in Rn , which of the following expressions does not make mathematical sense?

(A) (u + v) × w (B) u × v (w · z) (C) u × v × w (D) (u × v) · w

10. What is the area of the triangle defined by the points 1, 3, 2 , 2, 3, 1 , and 2, 2, 3 ? √ √ (A) 12 6 (B) 6 (C) 0 (D) 12 Free Response Questions

−→ 4, −3, 6 1. Find the terminal point Q of a nonzero vector u = P Q with initial point P

such that u is oppositely directed to v = 1, 1, 7 .

2. Let P be the point 1, 3, 0 and Q be the point 5, 9, 4 . Find the midpoint of the line segment between these two points.

3. Find a, b, and c if 4, b, c = a 2, 3, 2 .

Chapter 3

Elementary Linear Algebra 12e

–3–

Anton/Rorres

4. Find the scalars c1 , c2 , and c3 for which the following equation is satisfied.

c1 1, 1, 1 + c2 1, 1, 0 + c3 1, 0, 0 = 2, −3, 4

5. Let u = 1, 0, −2 , v = 1, 1, 1 , and w = 2, −2, 1 . Calculate u + v − 2 w .

6. Find the angle between the vectors 2, 3, 1 and 4, 1, 2 .

7. If u = 1, k, −2, 5 and u = 6, what is the value of k?

8. Prove that if u and v are unit vectors, then −1 ≤ u · v ≤ 1.

9. Prove using the definition of the norm that for vectors u and v in Rn , u − v = v − u .

10. Calculate the distance between the point 2, 1 and the line x − 2y + 3 = 0.

11. Calculate the distance between the point 3, 1, −2 and the plane 2x + y − z + 1 = 0.

12. Let u, v, and w be vectors in Rn . Prove that if u is orthogonal to both v and w, then u is orthogonal to the vector c1 v + c2 w for any scalars c1 and c2 .

13. Find proja u where a = 2, 5, 4 and u = 1, −2, 3 .

14. Find an equation for the plane containing the point 1, 1, 3 that is perpendicular to the line x = 2 − 3t, y = 1 + t, z = 2t.

15. Find a vector perpendicular to the line x, y, z = 1, 3, 2 + t 0, 1, −4 .

16. Find the vector equation of the line which contains the point 1, −5, 2 and is perpendicular to the plane 3x − 7y + 4z − 5 = 0.

17. Consider the system Ax = b. Prove that if x1 is a solution to Ax = b and x1 − x2 is a solution to the corresponding homogeneous system, then x2 is also a solution to Ax = b.

18. Let u = 1, 1, 3 , v = 2, 0, 1 , and w = 0, −1, 5 . Compute u × (v + w) .

19. Calculate the volume of the parallelepiped

defined by the vectors u = 2, 0, 5 , v = 1, −1, 0 , and w = 1, 3, 8 .

20. A plane contains the vectors u = 1, 3, −4 and v = 4, 1, −2 . Find a vector normal to the plane.

21. Let u and v be vectors in R3 . Prove that u × v 2 = (u · u)(v · v) − (u · v)2 .

Chapter 3

Elementary Linear Algebra 12e

–4–

Answers Multiple Choice Answers 1. (B) 2. (D) 3. (D) 4. (B) 5. (C) 6. (A) 7. (A) 8. (B) 9. (C) 10. (A) Free Response Answers

1. Possible answer: Q = 3, −4, −1

2. 3, 6, 2 3. a = 2, b = 6, c = 4 4. c1 = 4, c2 = −7, c3 = 5 5.

√

6−6

6. 40.696◦ √ 7. k = ± 6 10. √35 11. √106

Chapter 3

Anton/Rorres

Elementary Linear Algebra 12e 13.

8

, 4 , 16 45 9 45

–5–

14. −3(x − 1) + (y − 1) + 2(z − 3) = 0

15. Possible answer: 0, 4, 1

16. x = 1, −5, 2 + t 3, −7, 4 √ 18. 3 10 19. 4

20. Possible answer: −2, −14, −11 or 2, 14, 11

Chapter 3

Anton/Rorres

Chapter 4: General Vector Spaces Multiple Choice Questions 1. Let V = R2 and define addition and scalar multiplication as follows:

u + v = u 1 , u2 + v 1 , v 2 = u 1 + v 1 , 0

ku = k u1 , u2 = ku1 , 0 Which of the following vector space axioms does not hold?

(A) k(u + v) = ku + kv (B) 1u = u (C) Closure under scalar multiplication.

(D) None of the above.

2. Let V be a vector space, let k be a scalar, and let u, v, w be vectors in V . Which of the following statements does not hold?

(A) (u + v) − w = u − (w − v) (B) If ku = 0, then k = 0 or u = 0.

(C) −k(u + v − w) = −(ku − k(v − w)) (D) 0 − k(1u − v) + 0w = k(1v − u) + 0 3. Which of the following is a subspace of R3 ?

(A) All vectors of the form 0, a, a2 .

(B) All vectors of the form a + 2, a, 0 .

(C) All vectors of the form a, b, 2 .

(D) All vectors of the form a, b, a − 2b .

4. Which of the following is not a linear combination of A and B?

1 4 2 0 A= , B= 1 2 3 0

2 8 3 −4 3 4 1 12 (D) (C) (B) (A) 6 −1 −1 4 4 2 8 −2

Elementary Linear Algebra 12e

–2–

Anton/Rorres

5. Which of the following sets of vectors does not span R3 ?

(A) 2, 3, 4 , 6, 0, 7 , 0, 9, 5

(B) 1, 3, 2 , 1, 0, 1 , 0, 2, 2

(C) 1, 4, 6 , 2, 3, 1 , 1, 1, 0

(D) 6, 7, 1 , 2, 2, 3 , 1, 1, 2 6. Which of the following vectors in R3 does not lie on the same line as the others?

(A) 1, −7, 4 (B) −4, −28, 16 (C) −3, 21, −12 (D) 2, −14, 8 7. Which of the following sets of matrices are linearly independent?

2 3 4 0 3 1 (A) , , 1 0 1 0 −1 1

1 0 0 1 1 −1 , , (B) 2 2 0 −1 2 1

0 2 1 1 1 3 (C) , , 2 1 1 1 3 2

1 0 1 1 −1 1 , (D) , 2 1 5 2 1 0 8. The following set forms a basis for M22 . Which of the following is a possible value for the matrix A?

1 1 1 1 1 0 , , ,A 1 0 0 1 1 1

0 0 2 1 −1 −1 0 12 (D) (B) (C) 1 1 (A) 2 −2 1 2 0 −1 2 2 9. Given that the set {1, 1 + x + x2 , p(x)} is a basis for P2 , which of the following is a possible value for p(x)?

(A) 0 (B) 1 + x (C) −1 (D) 2 + x + x2

Chapter 4

Elementary Linear Algebra 12e

–3–

Anton/Rorres

10. Which of the following sets in R3 is linearly dependent?

(A) 1, 4, 6 , 1, −4, 0 , 4, 5, 2 , 1, 3, −5

(B) 3, 2, 4 , 2, 4, 3 , 0, 1, 3

(C) 0, 2, 0 , 2, 3, 3 , 4, 2, 4

(D) 1, 4, −2 , 3, 0, 0 11. How many vectors are in the standard basis for the vector space P5 ?

(A) 4

(B) 5

(C) 6

(D) 0

12. What is the transition matrix

from S1 to S2 given S1 = {u1 = 1, −2 , u2 = 3, −4 } and S2 = {v1 = 1, 3 , v2 = 3, 8 }?

−14 −36 (A) 5 13

− 13 −18 2 (B) 5 7 2

−7 −18 (C) 13 5

(D)

2

2

−13

−36

5

14

13. Given the basis B = vector [w]B ?

(A) 12 , 3

(B) 8, 3

(C) 2, 6

(D) 2, 3

4, 0 , 0, 1 and the vector w = 2, 3 , what is the coordinate

Chapter 4

Elementary Linear Algebra 12e

–4–

Anton/Rorres

14. Which is equivalent to the following matrix multiplication? ⎤⎡ ⎤ ⎡ 1 2 2 0 ⎥⎢ ⎥ ⎢ ⎢0 4 2 ⎥ ⎢3 ⎥ ⎦⎣ ⎦ ⎣ 5 5 4 2 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 0 2 2 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ (A) 1 ⎢ ⎣ 0⎦ + 3 ⎣ 4 ⎦ + 5 ⎣ 2⎦ 2 4 5 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 2 0 5 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ (B) 1 ⎢ ⎣ 2⎦ + 3 ⎣ 4 ⎦ + 5 ⎣ 4⎦ 0 2 2 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 1 3 5 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ (C) 2 ⎢ ⎣ 0⎦ + 2 ⎣ 4 ⎦ + 0 ⎣ 2⎦ 5 4 2 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 2 2 0 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ (D) 1 ⎢ ⎣ 2⎦ + 3 ⎣ 4 ⎦ + 5 ⎣ 4⎦ 5 0 2 15. Which vector is not in the column space of the following matrix? ⎡ ⎤ 9 4 14 ⎢ ⎥ ⎢4 1 ⎥ 7 ⎣ ⎦ 6 6 6 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 13 8 5 12 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ (A) ⎢ ⎣ 2⎦ (B) ⎣ 5⎦ (C) ⎣ 7⎦ (D) ⎣3⎦ 12 12 0 6 16. If rank(A) = 4 and nullity(AT ) = 2, which of the following is a possible dimension of A? (A) 6 × 5 (B) 6 × 3 (C) 3 × 6 (D) 5 × 6

Chapter 4

Elementary Linear Algebra 12e

–5–

Anton/Rorres

17. If A is an invertible n × n matrix, which of the following is not true?

(A) Ax = 0 has only the trivial solution.

(B) The column vectors of A span Rn .

(C) rank(AT ) = 0 (D) A has nullity 0.

Free Response Questions 1. Let V = R2 , and define addition and scalar multiplication as follows:

u + v = u 1 , u2 + v 1 , v 2 = u 1 v 1 , u2 v 2

ku = k u1 , u2 = uk1 , uk2 Given that V is a vector space, (a) determine the value of 0 in V. (b) verify that k0 = 0 holds.

(c) let u = u1 , u2 be in V . Find the value of −u such that u + (−u) = 0. (d) verify that (−1)u = −u holds.

2. Let V = R2 , and define addition and scalar multiplication as follows:

u + v = u1 , u2 + v1 , v2 = u1 + v1 + 1, u2 + v2 + 2

ku = k u1 , u2 = ku1 , −ku2

Let u = 2, 1 and v = 4, −1 . Compute the value of 2u + v under the given operations. (Note that V is not a vector space.)

3. Let V be the set of all functions of the form sin (cx) where c is a real number. Define addition and scalar multiplication to be: sin (c1 x) + sin (c2 x) = sin ((c1 + c2 )x) k sin (cx) = sin (kcx) Prove that V is a vector space under the given operations.

4. Let V be the set of pairs of real numbers x, y such that x ≥ 0 and y ≥ 0. Define addition and scalar multiplication to be the standard operations in R2 . Prove that the following vector space axioms do not hold. (a) For each u in V , there is an object −u in V , such that u + (−u) = (−u) + u = 0. (b) If k is any scalar and u is any object in V , then ku is in V .

Chapter 4

Elementary Linear Algebra 12e

–6–

Anton/Rorres

5. Determine whether the set of all polynomials in the form a0 + a1 x + a2 x2 where a0 , a1 , and a2 are integers is a subspace of P3 . Justify your answer.

6. Express 22, 57, −6 as a linear combination of vectors a = 4, 9, 3 and b = −2, −7, 6 .

7. Let W be the subset of M33 of “magic squares”, i.e. 3 × 3 matrices such that for some number c, the sum of any row = the sum of any column = the sum of any diagonal = c.

Prove that W is a subspace of M33 .

8. Let W be the subset of R3 such that for x, y, z in W , x2 + y 2 = z 2 . Show that W is closed under scalar multiplication but not under addition.

9. Determine whether the vectors x2 − 1, x2 + x − 2, and x2 + 3x + 2 are linearly independent or linearly dependent in P2 .

10. Are the vectors 1, 2, 4, −3 , 1, 1, 0, 1 , and 2, 1, 1, 3 in R4 linearly independent or linearly dependent?

11. By using appropriate identities, prove that the vectors f (t) = sin (t) , g(t) = cos (t) , h(t) = t in F (−∞, ∞) are linearly independent.

12. Prove that if {v1 , v2 , v3 } is a linearly independent set of vectors in some vector space V , then {v1 + v2 , v1 − v2 , v1 − 2v2 + v3 } is also a linearly independent set in V .

13. Use a determinant to prove that the vectors 6, 3, 2 , 3, 6, 0 , and 0, 0, 2 form a basis for R3 .

8, −4, −12 14. Determine the coordinate vector of w = relative to the basis

S = 1, 1, 1 , 1, 5, −3 , 2, 2, 1 .

15. Let {v1 , v2 , . . . , vn } be a basis for a vector space V , and let w be a vector in V . Prove that the set {w, v1 , v2 , . . . , vn } (a) is linearly dependent. (b) spans V .

16. Show that the following vectors form a basis for P4 .

1, (t − 1), (t − 1)2 , (t − 1)3 , (t − 1)4

Chapter 4

Elementary Linear Algebra 12e

–7–

Anton/Rorres

17. Determine a basis for the solution space of the system, and find the dimension of the solution space.

2x1 + 5x2 + x3 = 0 x1 + 3x2 + 2x3 = 0 3x1 + 4x2 − 9x3 = 0 18. Determine a basis for the vector space defined by the set of all matrices in the form

a b c d

e

f

and determine the dimension of the space.

19. Find a basis for the subspace W of M22 spanned by

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

1

0

20. Let V be a vector space with dimension n. Prove that any set with less than n vectors in V cannot span V .

21. Let V be the set of symmetric 2 × 2 matrices. Prove that dim(V ) = 3.

22. The following matrix Q is the transition matrix from the standard basis S = {e1 , e2 , e3 } to what basis B? ⎡ ⎤ 5 4 1 ⎢ ⎥ ⎢ 2 3 4⎥ ⎣ ⎦ 3 2 3 23. Find a transition matrix which changes vectors in R3 from the form (a, b, c) to (a, c, b).

24. Consider the basis B = {u1 , u2 , u3 } for R3 where ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 1 1 0 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ u1 = ⎢ ⎣2⎦ , u2 = ⎣3⎦ , u3 = ⎣1⎦ 0 2 3 ⎡ ⎤ a ⎢ ⎥ ⎥ Compute [w]B for w = ⎢ ⎣ b⎦ . c

Chapter 4

Elementary Linear Algebra 12e

–8–

Anton/Rorres

25. Let S = {e1 , e2 , e3 } be the standard basis for R3 , and let B = {v1 , v2 , v3 } be the basis that results when the linear transformation defined by

T x1 , x2 , x3 = 2x2 + x3 , x1 − 4x2 , 3x1 is applied to each vector in S. Find the transition matrix PB→S .

26. Find a basis for the row space of the following matrix. ⎤ ⎡ 6 4 0 ⎥ ⎢ ⎢0 4 −1⎥ ⎦ ⎣ 6 −8 3 27. Find a basis for the null space of the following matrix. ⎡ ⎤ 1 −3 3 ⎢ ⎥ ⎢2 −6 ⎥ 8 ⎣ ⎦ 3 −9 11 28. Prove that the column vectors of an n × n invertible matrix form a basis for Rn .

29. Let A and B be n × n invertible matrices. Prove that if a vector b is in the column space of B, then the vector Ab is in the column space of the matrix AB. 30. Find the rank and nullity of the following matrix. ⎡ ⎤ 1 1 7 ⎢ ⎥ ⎢1 2 ⎥ 0 ⎣ ⎦ 2 3 10 31. Show that rank(A) = rank(AT ) for the following matrix. ⎡ ⎤ 1 −4 5 1 ⎢ ⎥ ⎢ 2 −8 9 0⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 1 −4 ⎥ 3 −3 ⎣ ⎦ −1 4 −2 5 32. Let A be an m × n matrix. Prove that the set containing both the null space of A and the row space of A forms a basis for Rn .

33. Prove that a square n × n matrix A is invertible if and only if rank(A) = n.

Chapter 4

Elementary Linear Algebra 12e

–9–

Answers Multiple Choice Answers 1. (B) 2. (C) 3. (D) 4. (D) 5. (A) 6. (B) 7. (A) 8. (C) 9. (B) 10. (A) 11. (C) 12. (A) 13. (A) 14. (A) 15. (C) 16. (A) 17. (C) Free Response Answers

1. (a) 0 = 1, 1

(b) k0 = k 1, 1 = 1k , 1k = 1, 1 = 0

(c) −u = u11 , u12

1 1 −1 (d) (−1)u = (−1) u1 , u2 = u−1 = u1 , u2 = −u 1 , u2

Chapter 4

Anton/Rorres

Elementary Linear Algebra 12e

– 10 –

Anton/Rorres

2. 9, −1 5. No. Closure under scalar multiplication fails.

6. 4a − 3b 9. Independent 10. Independent

14. −53, −3, 32 17. basis: {(7, −3, 1)}; dimension: 1

1 0 0 0 1 0 0 0 18. basis: , , 0 0 0 0 0 0 0 0 dimension: 6

1 1 1 0 1 19. Possible solution: , , 1 1 0 1 0

22. B = ⎡ 1 ⎢ 23. ⎢ ⎣0 0 ⎡

0 0 1

1

,

0

1

1

0

0

0

⎤

4a − 2b + c ⎢ 25. ⎢ ⎣1 3

0

1 1 5 3 7 5 1 1 13 , − , , , , , − , − , 24 4 24 12 2 12 24 4 24 ⎤ 0 ⎥ 1⎥ ⎦ 0

7a − 3b + c

0

0

⎥ ⎢ ⎥ 24. ⎢ ⎣−6a + 3b − c⎦ ⎡

0

2

1

⎤

−4

⎥ 0⎥ ⎦

0

0

3, 2, 0 , 0, 4, −1

27. Possible answer: 3, 1, 0

26. Possible answer:

30. rank = 3, nullity = 0 31. rank(A) = rank(AT ) = 2

Chapter 4

,

0

0

0

0

1

0

,

0

0

0

0

0

1

Chapter 5: Eigenvalues and Eigenvectors Multiple Choice Questions 1. Which of the following is not an eigenvector of the matrix

(A)

0 1

(B)

2 2

(C)

0 2

(D)

3

0

1

9

?

−6 1

2. What is the sum of the dimensions of the eigenspaces of the following matrix? ⎡ ⎤ 2 0 0 0 ⎢ ⎥ ⎢ 0 2 0 0⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 0 2 2⎥ ⎣ ⎦ 0 0 2 3 (A) 1

(B) 2

(C) 4

(D) 8

1 1 ?

3. Which matrix is similar to 1 1

2 2 0 −3 1 (A) (B) (C) 1 1 −3 1 1

−3 0

(D)

−1

−3

1

3

4. What is the value of the term in row 1, column 1 of the matrix A5 where A = (A) 636

(B) 0

(C) 243

(D) 905

5. Given u = 7, 3 + i, 4 − 6i , what is Re(u)?

(A) 0, 1, −6 (B) 7, 3 + i, 4 − 6i (C) 7, 3, 4 (D) 7, 3 − i, 4 + 6i 6. Which of the following expressions is not equivalent to ku · ( v + w )?

(A) k u · ( v + w ) (B) ku · ( v + w ) (C) ku · ( v + w ) (D) k u · ( v + w )

3

1

1

4

?

Elementary Linear Algebra 12e

–2–

Anton/Rorres

7. Which of the following is a solution to the system y =

(A) y = c1 e (B) y = c1 e (C) y = c1 e (D) y = c1 e

1 −4t 1

−4 t 0

1 −4t 0

1 −4t 0

+ c2 e + c2 e

−4

2

0

−2

1 −2t

y?

1

2 t

+ c2 e + c2 e

−2

1 2t −1

1 −2t 1

8. The solution to the system y − 5y + 6y = 0 with initial conditions y(0) = 3, y (0) = 7 is of the form y = c1 eλ1 t + c2 eλ2 t . What are the values of the constants c1 and c2 ?

(A) c1 = 1, c2 = 2 (B) c1 = 5, c2 = −2 (C) c1 = 53 , c2 = 43 (D) c1 = −1, c2 = 4 9. Which of the following is a stochastic matrix? ⎡ ⎤ ⎡ ⎤ ⎡ 0.1 0.2 0.8 0 −1 1 0.2 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ (A) ⎢ 2 0⎥ ⎣0.1 0.2 0.2⎦ (B) ⎣0 ⎦ (C) ⎣0.1 0.8 0.6 0.2 1 0 0 0.4

⎤ 0.3

0.4

⎡ 0.2

0.3

⎥ 0.7⎥ ⎦

⎢ (D) ⎢ ⎣0.7

0.6

0.2

0.8

0.1

0.3

0.1

⎤ 0.9 ⎥ 0⎥ ⎦ 0.1

10. Let P be a transition matrix and q be its steady state vector. What is the eigenvalue associated with q?

(A) λ = 0

(B) λ = 1

(C) λ = −1

(D) none of the above

Chapter 5

Elementary Linear Algebra 12e Free Response Questions ⎡ −9 −6 ⎢ 1. Let A = ⎢ 2 ⎣ 1 4 2

–3–

−22

Anton/Rorres

⎤

⎥ 2⎥ ⎦.

10

(a) Find the characteristic equation of A. (b) Find the eigenvalues of A. (c) Find bases for the eigenspaces of A. 2. Given that the characteristic polynomial of a matrix A is p(λ) = (λ + 1)(λ − 2)2 (λ + 3)2 , find det (A−1 ).

3. Prove that if λ is an eigenvalue of A, then λk is an eigenvalue of kA−1 for every scalar k.

4. Prove that if v1 , v2 are eigenvectors of a matrix A corresponding to the eigenvalue λ, then the vector v = av1 + bv2 is an eigenvector of A with corresponding eigenvalue λ for every scalar a and b.

5 2 5. Find a matrix P that diagonalizes A = and determine P −1 AP .

2 2

6. Find An if n is a positive integer and A =

4

−5

−3

2

.

7. Prove that if A and B are similar matrices, then if λ is an eigenvalue of A it is also an eigenvalue of B. 8. Let A, B, and C be matrices. Prove that if A and B are similar, and B and C are similar, then A and C are similar.

9. Find the eigenvalues and bases for the eigenspaces of the following matrix.

6 −13 1

0

10. Compute 1, 4 + i, 3 − 2i · 2, −3i, 4 + 2i .

11. Let A be a real matrix. Prove that if v is an eigenvector of A with corresponding eigenvalue λ, then v is also an eigenvector of A with corresponding eigenvalue λ.

12. Let A be a real matrix with real and complex eigenvalues λ1 , λ2 , ..., λn . Use the fact that det (A) = λ1 λ2 · · · λn to prove that det (A) is always a real number.

Chapter 5

Elementary Linear Algebra 12e

–4–

Anton/Rorres

13. Let the solution to some system of equations in y1 and y2 be y1 = 2c1 e3t + c2 et and y2 = 4c1 e3t + 5c2 et . Given the initial conditions y1 (0) = 1 and y2 (0) = 0, find the values of c1 and c2 .

14. Find the solution to y = Ay if the eigenvalues of A are λ1 = 2 and λ2 = 3 with 1 −1 and v2 = ? corresponding eigenvectors v1 = 0 1 15. Prove that if y1 = eλ1 x x1 and y2 = eλ2 x x2 are solutions to the system y = Ay, then c1 y1 + c2 y2 is also a solution for real scalars c1 and c2 .

2 6 16. Let A = , let P diagonalize A, and let D = P −1 AP . Show by finding a coun0 1 terexample that the two systems y = Ay and u = Du do not have the same solutions.

17. Fill in the missing entries of the stochastic matrix. ⎡ ⎤ 1 5 1 0 10 4 9⎥ ⎢ ⎢3 1 1 2⎥ ⎢ 2 9⎥ P = ⎢ 85 10 ⎥ 3 1 2⎥ ⎢ ⎣ 8 10 8 9 ⎦ ∗ ∗ ∗ ∗ 18. In a certain region, there are people who live in the city and people who live in the country. Records show that every year 90% of people living in the country stay in the country, and 50% of people living in the city move to the country. Initially, 70% of people in the region live in the city and the rest live in the country. (a) Find a transition matrix for this scenario. (b) What percentage of the region will live in the country in 1 year? (c) If there are 1,000,000 people living in the region, how many people will be living in the country and the city in 2 years?

19. Find the value of p in the following probability vector. ⎡ ⎤ 2p ⎢ ⎥ ⎥ x=⎢ ⎣ p⎦ 3p

0.7 0.1 be a transition matrix. Find the steady state vector of the Markov 20. Let P = 0.3 0.9 Chain.

Chapter 5

Elementary Linear Algebra 12e

–5–

Anton/Rorres

Answers Multiple Choice Answers 1. (B) 2. (C) 3. (D) 4. (A) 5. (C) 6. (D) 7. (D) 8. (A) 9. (D) 10. (B) Free Response Answers 1. (a) λ3 − 3λ2 + 2λ = 0 (b) λ = 0, λ = 1, λ = 2, ⎡

−8

⎤ ⎡

−5

⎤ ⎡

−2

⎤

⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ (c) ⎢ ⎣ 1 ⎦ , ⎣ 1⎦ , ⎣ 0⎦ 1 2 3 1 2. − 36

5. Possible answer: P =

−1

2

2

1

5(7)n + 3(−1)n 1 6. An = 8 −3(7)n + 3(−1)n

, P −1 AP =

−5(7)n + 5(−1)n

1

0

0

6

3(7)n + 5(−1)n

Chapter 5

Elementary Linear Algebra 12e

–6–

9. λ = 3 ± 2i, v1 = 3 + 2i, 1 , v2 = 3 − 2i, 1 10. 21 − 14i 13. c1 = 56 , c2 = − 23 14. y = c1 e

1 2t 0

+ c2 e

−1 3t 1

16. Possible Answer: u =

c1 e2x c2 e

x

is a solution to u = Du but u = Au.

17. 0, 12 , 18 , 0 18. (a) Possible answer:

0.9

0.5

0.1

0.5

(b) 62% (c) 748,000 in the country, 252,000 in the city 19. p = 16

20. q =

1 4 3 4

Chapter 5

Anton/Rorres

Chapter 6: Inner Product Spaces Multiple Choice Questions

1. Given u = u1 , u2 and v = v1 , v2 , which of the following operations defines an inner product on R2 ?

(A) u, v = u1 v2 + u2 v1

(B) u, v = 3u1 v1 − 2u2 v2

(C) u, v = u1 v1 + 2u2 v2

(D) u, v = u1 v2 + 2u2 v1 2. Which of the following is not true?

(A) k v = |k| v (B) d(u, v) = d(v, u)

(C) u + v, w = u, w + v, w

(D) ku, kv = k u, v

3. Given u, v = −5, which of the following is a possible value for u v ?

(A) 4

(B) 6

(C) −6

(D) −4

4. Which of the following vectors in R2 are orthogonal with respect to the Euclidean inner product?

(A) 1, 2 and −3, 2

(B) 4, 6 and 3, 2

(C) 4, 6 and −3, 2

(D) 1, 2 and 3, 2 5. Which of the following sets of vectors in R2 are orthonormal with respect to the Euclidean inner product?

(A) 1, 0 , 0, 1

(B) 2, 0 , 0, 2 √ √ √ √ 2 2 , 2 2 (C) , , 2 2 2 2

(D) 1, 0 , 0, 0

Elementary Linear Algebra 12e

–2–

Anton/Rorres

6. Let {u1 , u2 , u3 } be a basis for R3 , and let {v1 , v2 , v3 } be the orthonormal set constructed via the Gram-Schmidt process. If v1 = √13 , √13 , √13 , v2 = √16, , √16 , − √26 , and u3 =

1, 0, 0 , what is the value of v3 ?

(A) 32 , 12 , 0 (B) (C) (D)

√ √ 3 10 10 , , 0 10 10

1

, − 12 , 0 2

√

√ 2 2 , − , 0 2 2

1, 3, −2 7. Which of the following vectors is the orthogonal projection of on the subspace

of R3 spanned by 1, 0, 3 , 1, 1, 2 ?

8 34 , 11 , − 10 (A) 11 11

5 35 , 11 , − 20 (B) 11 11

(C) −85, −35, −220

(D) −8, −2, −22 8. Which system is the associated normal equation to the following system? ⎡ ⎤ ⎡ ⎤ 5 1 0 ⎢ ⎥ ⎢ ⎥ x1 ⎥ ⎢1 2 ⎥ =⎢ ⎣4 ⎦ ⎣ ⎦ x2 4 5 3

18 8 2 x1 = (A) 6 2 7 x2

27 17 x1 29 (B) = 17 13 x2 20

8 9 1 x1 = (C) 2 1 0 x2

20 1 0 x1 = (D) 20 0 1 x2 Free Response Questions

1 1. Given A = −2 product on M22 .

3 4

and B =

2

−1

2

0

, calculate A, B using the standard inner

Chapter 6

Elementary Linear Algebra 12e

–3–

Anton/Rorres

2. Let u = 1, −3 and v = 5, 1 . Use the inner product u, v = 3u1 v1 + 5u2 v2 on R2 to compute d(u, v).

3. Prove that

u + v 2 = u 2 + 2 u, v + v 2

where u and v are vectors in some inner product space.

4. Prove that

u + v, u − v = u 2 − v 2

where u and v are vectors in some inner product space.

5. If u = 4, v = 2, −1, −2 and u and v are orthogonal, find u + v .

6. Compute the angle between the following pair of vectors in M22 using the standard inner product on M22 .

1 0 1 0 A= , B= 0 1 1 0 7. Let W be a subspace of a real inner product space V . Prove that if u and v are vectors in W ⊥ , then for scalars c1 , c2 , the vector c1 u + c2 v is also in W ⊥ .

8. Let V be a real inner product space. Prove that V ⊥ = {0}.

9. Let u =

√1 , √1 8 8

and v = 12 , − 14 . Using the inner product u, v = 3u1 v1 + 5u2 v2 on

R2 , determine whether the set {u, v} is (a) orthogonal. (b) normal.

10. Use the Gram-Schmidt process to transform the basis 1, 2, −1 , 1, 3, 0 , 4, 1, 0 into an orthonormal basis under the Euclidean inner product in R3 .

11. Let {v1 , v2 , v3 } be the basis formed by performing the Gram-Schmidt process on the basis {u1 , u2 , u3 }. Prove that the vectors v1 , v2 , and v3 are orthogonal.

12. Let Q be an n × n matrix whose column vectors form an orthonormal basis for Rn .

Prove that Q is invertible and that Q−1 = QT .

13. Use det(AT A) to determine whether the column vectors of the following matrix are linearly independent. ⎡ ⎤ 1 2 1 ⎢ ⎥ ⎢0 ⎥ 1 −1 ⎢ ⎥ A=⎢ ⎥ ⎢1 −1 0⎥ ⎣ ⎦ 1 −1 0

Chapter 6

Elementary Linear Algebra 12e

–4–

Anton/Rorres

14. Determine the least squares solution to the following system. ⎤ ⎡ ⎤ ⎡ 3 7 6 ⎥ x1 ⎢ ⎥ ⎢ ⎥ ⎢ 0 3⎥ =⎢ ⎦ ⎣12⎦ ⎣ x2 10 −2 1 15. Let A be an m × n matrix. Prove that the matrices AT A and AAT are both square.

16. Let A be an n × n matrix and let x be a vector in Rn . Prove that the system AT Ax = b is consistent for every b in Rn if and only if the system Ax = b is consistent for every b in Rn .

0, 2 1, 1 17.

Find the least squares straight line fit y = a x + a to the data points , , 0 1

2, 3 , and 3, 3 .

2 0, 0 0, 1 18.

Find the least squares quadratic fit y = a + a x + ax to the data points , , 0 1 2

1, 2 , 1, 3 , and 2, 0 .

19. Find the least squares approximation of f (x) = ex over the interval [0, 2π] by a trigonometric polynomial of order 3.

20. Find the least squares approximation of f (x) = x3 over the interval [0, 2π] by a trigonometric polynomial of order 2 or less.

Chapter 6

Elementary Linear Algebra 12e

–5–

Anton/Rorres

Answers Multiple Choice Answers 1. (C) 2. (D) 3. (B) 4. (C) 5. (A) 6. (D) 7. (B) 8. (B) Free Response Answers 1. −5 √ 2. 8 2 5. 5 π 6.

3 9. (a) not orthogonal (b) not normal

√1 , √2 , − √1 , − √1 , √4 , √7 10. , √311 , − √111 , √111 6 6 6 66 66 66 13. det(AT A) = 32 = 0, therefore the columns of A are linearly independent.

−3 14.

4 17. 32 + 12 x 18. 12 + 17 x − 94 x2 4 2π

19. f (x) ≈ e π−1

1 2

1 3 + 12 cos (x) − 12 sin (x) + 15 cos (2x) − 25 sin (2x) + 10 cos (3x) − 10 sin (3x)

20. f (x) ≈ 2π 3 + 12πcos (x) + 3πcos (2x) + (12 − 8π 2 )sin (x) + ( 32 − 4π 2 )sin (2x)

Chapter 6

Chapter 7: Diagonalization and Quadratic Forms Multiple Choice Questions 1. Which of the following matrices is orthogonal? ⎡ ⎤ ⎡ ⎤ ⎡ 0 0 0 1 2 1 0 ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ ⎢ (A) ⎢ 0 0⎥ 1 2⎥ ⎣1 ⎦ (B) ⎣0 ⎦ (C) ⎣1 0 −1 0 1 −2 1 0

0 0 1

⎤

√1 2⎥

0⎥ ⎦

√1 2

⎡

3 5

− 35

0

⎢ (D) ⎢ ⎣0

⎥

4⎥ 5⎦

− 35 4 5

4 5

⎤

0

2. Which of the following is not true?

(A) If A is orthogonal, then |det (A)| = 1.

(B) If A and B are orthogonal matrices, then AB is also orthogonal.

(C) If A is orthogonal, then kA is also orthogonal for any scalar k.

(D) If A is orthogonal, then (A−1 )T is also orthogonal.

3. Let A be an n × n matrix. Which of the following statements is not equivalent to the others?

(A) A is orthogonally diagonalizable.

(B) A has determinant ±1.

(C) A has an orthonormal set of n eigenvectors.

(D) A is symmetric.

4. What is the characteristic polynomial of the following symmetric matrix?

4 3 3 (A) λ2 + 4λ + 9

(B) λ2 − 25

−4

(C) λ2 − 7

(D) 4λ2 + 9

5. Which of the following is not a quadratic form?

(A) x2 − 5y 2 + 4x + y √ (B) (2x1 + 2x2 )2 (C) 3x21 + 5x22 − 7x23 − 5x1 x2 − 6x2 x3 (D) 14x1 x3 − x23 6. Which of the following matrices is negative definite? ⎤ ⎤ ⎡ ⎡

−1 0 0 3 0 0 ⎥ ⎥ ⎢ ⎢ 3 ⎢ (A) ⎢ 0⎥ 0⎥ ⎦ (C) ⎦ (B) ⎣ 0 1 ⎣0 0 −1 0 0 −1 0 0 −17

−1 3

(D)

−6

1

1

−6

Elementary Linear Algebra 12e

–2–

Anton/Rorres

7. The Hessian of a function f is given by H x, y =

48x

0

0

6y

. Which of the following

characterizes f at the critical point x0 , y0 = − 12 , −2 ?

(A) Cannot be determined from the given information.

(B) f has a saddle point at x0 , y0 .

(C) f has a relative maximum at x0 , y0 .

(D) f has a relative minimum at x0 , y0 .

8. Let λ1 and λ2 be the eigenvalues of the Hessian matrix for a function f at a critical point x0 , y0 . If λ1 < 0, which of the following statements cannot be true?

(A) f has a saddle point at x0 , y0 .

(B) f has a relative minimum at x0 , y0 .

(C) f has a relative maximum at x0 , y0 .

(D) f = 0 at x0 , y0 .

9. Let A and B be n × n complex matrices, and let k be a complex scalar. Which of the following is not always true?

(A) (A∗ )∗ = A (B) (A + B)∗ = A∗ + B ∗ (C) (kA)∗ = kA∗ (D) (AB)∗ = B ∗ A∗ 10. What value of k makes the following matrix Hermitian? ⎤ ⎡ 5 2 + i 1 − 11i ⎥ ⎢ ⎥ ⎢2 − i 12 0 ⎦ ⎣ k 0 0 (A) 1 − 11i

(B) 1 + 11i

(C) −1 − 11i

(D) −1 + 11i

Free Response Questions 1. Show that the following matrix is orthogonal by showing that the row vectors of the matrix form an orthonormal set in Rn with the Euclidean inner product. ⎤ ⎡ 1 ⎢3 ⎢2 ⎣3 2 3

2 3 1 3 − 23

2 3 ⎥ − 23 ⎥ ⎦ 1 3

Chapter 7

Elementary Linear Algebra 12e

–3–

Anton/Rorres

2. Given that the following matrix is orthogonal, what is the value of a?

a √310 − √310

3. Prove that

cos (θ)

sin (θ)

−sin (θ)

cos (θ)

√1 10

is orthogonal for any real θ.

4. Let A be an orthogonal matrix. Prove that for any positive integer k, Ak is also orthogonal.

5. Let A be an orthogonal matrix, and let λ be an eigenvalue of A. Prove that |λ| = 1.

6. Find a matrix P that orthogonally diagonalizes the following matrix A, and determine the value of D = P −1 AP . ⎡ ⎤ −1 2 0 ⎢ ⎥ ⎥ A=⎢ 2 0 2 ⎣ ⎦ 0 2 1 7. Find a symmetric 3 × 3 matrix whose eigenvalues are 8,−1, and −1 with corresponding eigenvectors 2, 1, 2 , −1, 0, 1 , and −1, 2, 0 .

8. Let a matrix A be orthogonally diagonalized by P and have characteristic equation (λ − 1)2 (λ + 2).

Find a possible value for the matrix P T AP .

1 0 is not orthogonally diagonalizable.

9. Show that the matrix 1 1 10. Express the quadratic form 2x2 − 4xy + y 2 in matrix notation xT Ax where A is a symmetric matrix.

11. Find the formula for the following quadratic form that does not use matrices. ⎡ ⎤⎡ ⎤ a 0 0 x ⎢ ⎥⎢ ⎥ ⎥⎢ ⎥ [x y z] ⎢ ⎣ 0 b 0⎦ ⎣ y ⎦ 0 0 c z 12. Use a change of variables to reduce the following quadratic to a sum of squares.

4x21 + 4x22 + 14x1 x2

Chapter 7

Elementary Linear Algebra 12e

–4–

13. Show by definition that the matrix A =

Anton/Rorres

2

−1

−1

2

is positive definite.

14. Find the points where the maximum and minimum values of the function f x, y = 4x2 + 5y 2 occur subject to the constraint x2 + y 2 = 2.

15. Find the critical points of f and classify them as relative maxima, relative minima, or saddle points. f x, y = x3 + 3x2 − 9x + y 3 − 12y 16. A function f has Hessian matrix ⎡

⎤

−2

0

⎢ H x, y, z = ⎢ ⎣ 0

−2

0

0

0

⎥ 0⎥ ⎦. −2

What can you say about the classification of the critical points of f ?

17. Let the Hessian of a function f be given by ⎡ x2 + 1 0 ⎢ H(x, y, z) = ⎢ 0 y2 + 1 ⎣ 0 0

0

⎤

⎥ 0⎥ ⎦

z2 + 1

Prove that any critical point of f is a relative minimum.

18. Determine whether A is unitary and, if so, determine its inverse.

A=

√1 2 1 √ − 2

√i 2 √i 2

19. Find a unitary matrix P that diagonalizes the following matrix and determine P −1 AP .

−3 2 + 2i A= 2 − 2i 4 20. Prove that if A is a normal matrix, then (I + A∗ )2 is also normal.

21. Let A and B be real matrices. Prove that M = A + iB is skew Hermitian if and only if A is skew symmetric and B is symmetric.

Chapter 7

Elementary Linear Algebra 12e

–5–

Anton/Rorres

Answers Multiple Choice Answers 1. (A) 2. (C) 3. (B) 4. (B) 5. (A) 6. (D) 7. (C) 8. (B) 9. (C) 10. (B) Free Response Answers 2. √110 ⎡

1 3 2 3 2 3

− 23

⎢ 1 6. Possible answer: P = ⎢ ⎣− 3 2 3

⎡

⎡

⎤ 3

⎢ 7. Possible answer: ⎢ ⎣2

2 0

⎥ 2⎥ ⎦

4

2

3

1

⎢ 8. Possible answer: ⎢ ⎣0

0 1

⎥ 0⎥ ⎦

0

0

−2

⎡

⎤

2 0 3 ⎥ ⎢ ⎢ − 23 ⎥ ⎦, D = ⎣0 1 0 3

4

0

⎤

Chapter 7

0

0

⎤

3

⎥ 0⎥ ⎦

0

−3

Elementary Linear Algebra 12e

10. [x y]

2

−2

−2

1

–6–

x y

11. ax2 + by 2 + cz 2 12. 11y12 − 3y22 , where y1 = √12 (x1 + x2 ) and y2 = √12 (−x1 + x2 ).

13. xT Ax = x21 + (x1 − x2 )2 + x22 > 0 for x = 0 √ 14. maximum values: x, y = 0, ± 2 √ minimum values: x, y = ± 2, 0 15. 1, 2 : relative min, 1, −2 : saddle point, −3, 2 : saddle point, −3, −2 : relative max 16. All critical points are relative maxima.

18. A is unitary and A−1 = A∗ =

19. P =

1+i √ 3 2 4 √ 3 2

−2−2i 3 1 3

√1 2 i − √2

and P −1 AP =

− √12 − √i2 5

0

0

−4

Chapter 7

Anton/Rorres

Chapter 8: General Linear Transformations Multiple Choice Questions

1 1. Consider the basis B = {v1 , v2 } for R2 , where v 1 = 1, 1 and v2 = −1, . Let 2 2 T : R → R be the linear operator such that T (v1 ) = 3, 5 and T (v2 ) = −2, 3 . What is the value of T (2, 1)?

(A) 2, 1

, 6 (B) 11 2 5 1 , 19 (C) 19

(D) 4, 13 2. Which of the following transformations T : R2 → R is a linear transformation?

(A) T (a, b) = 3

(B) T (a, b) = a, b · a, b (C) T (a, b) = a2 (D) T (a, b) = 3a + 2b 3. Let T : V → W be a linear transformation, with dim(V ) = m and dim(W ) = n. For which of the following conditions is T one-to-one?

(A) R(T ) = W and m = n (B) m > n (C) rank(T ) = m − 1 and n > m (D) nullity(T ) = m 4. For which of the following linear transformations is nullity(T ) = 4?

(A) T : R3 → R8 , rank(T ) = 2 (B) T : P3 → P3 , rank(T ) = 3 (C) T : M23 → M33 , rank(T ) = 6 (D) T : R5 → R2 , rank(T ) = 1 5. Which of the following vector spaces is not isomorphic to R16 ?

(A) R16 (B) P16 (C) M44 (D) M82

Elementary Linear Algebra 12e

–2–

Anton/Rorres

6. Which of the following definitions of T : R2 → R2 is an isomorphism?

T (A) x, y −−→ y, x + y

T

(B) x, y −−→ 0, xy

T

(C) x, y −−→ x + y, x + y

T

(D) x, y −−→ 2x, x 7. Let T : V → V be a one-to-one linear operator, and let B be a basis for V . Which of the following matrices is a possible value of [T ]B ? ⎡ ⎤ ⎡ ⎤

1 0 −1 −1 0 1 1 ⎢ ⎥ ⎢ ⎥ − 2 ⎢0 1⎥ (C) ⎢−1 ⎥ (D) 1 (B) (A) 12 0 −1 ⎣ ⎦ ⎣ ⎦ 0 − 12 2 0 0 0 −1 −1 8. Let T1 : U → V , T2 : V → W , and T3 : W → U . Let B, B , and B be bases for U , V , and W respectively. Which of the following expressions is not equivalent to [T3 ◦ T2 ◦ T1 ]B ?

(A) [T3 ]B,B [T2 ◦ T1 ]B ,B (B) [T3 ◦ T2 ]B,B [T1 ]B ,B (C) [T3 ]B ,B [T2 ]B ,B [T1 ]B,B (D) [T3 ]B,B [T2 ]B ,B [T1 ]B ,B 9. If A and B are similar matrices, which of the following is not necessarily true?

(A) rank(A) = rank(B) (B) tr (A) = tr (B) (C) A−1 and B −1 exist and are similar.

(D) Av = λv if and only if Bv = λv.

10. Let T : V → V be a linear operator and let B and B be bases for V . If [T ]B = which of the following is a possible value for [T ]B ?

2 −1 4 0 3 1 (C) (B) (A) −1 3 0 −1 1 2

(D)

−3

0

0

6

5

−3

2

−2

,

11. Which of the following operators does not have an inverse?

(A) T : R3 → R3 is the clockwise rotation about the negative y-axis through an angle θ.

(B) T : R2 → R2 is the contraction with factor k = 12 .

(C) T : R2 → R2 is the shear in the y-direction with factor k = − 13 .

(D) T : R3 → R3 is the orthogonal projection onto the xz-plane.

Chapter 8

Elementary Linear Algebra 12e Free Response Questions 1. Let T : M22 → R be defined as T

–3–

a

b

c

d

Anton/Rorres

= a − d. Show that T is a linear transfor-

mation and state its kernel.

2. Let the linear operator T : R2 → R2 be defined by T (x, y) = x − 2y, 2x + 3y . (a) Find a basis for the range. (b) Find a basis for the kernel. (c) Find the rank and nullity.

3. Let T : V → W be a transformation. Prove that if T (cv1 + v2 ) = cT (v1 ) + T (v2 ) for any vectors v1 and v2 in V and any scalar c, then T is a linear transformation.

4. Let T : V → W be a linear transformation. (a) Prove that the kernel of T is a subspace of V . (b) Prove that the range of T is a subspace of W .

5. Use whether the transformation T : M22 → M23 defined as

to determine the kernel a b 2 0 −1 a b is one-to-one. = T 2 c d 1 3 c d 6. Let T : P3 → P3 be defined as T (p(x)) = p(x − 3). Show that T is one-to-one and find a formula for T −1 (p(x)).

7. Let T : V → V be a one-to-one linear operator, and let dim(V ) = n be finite. (a) Prove that T −1 exists. (b) Prove that T −1 is one-to-one. (c) Prove that (T −1 )−1 = T .

8. Prove that the identity operator is one-to-one and that its inverse is the identity operator.

9. Let TA : R3 → R3 be an isomorphism with ⎡ 0 1 ⎢ A=⎢ ⎣1 k k 2

k

⎤

⎥ 2⎥ ⎦ 0

where k is real. Find all possible values for k.

10. Show that P3 is isomorphic to M22 by finding an isomorphism T : P3 → M22 .

11. Let U and V be vector spaces.

Chapter 8

Elementary Linear Algebra 12e

–4–

Anton/Rorres

(a) Prove that U is isomorphic to itself. (b) Prove that if U is isomorphic to V , then V is isomorphic to U .

12. Prove that if T1 : U → V and T2 : V → W are isomorphisms, then T2 ◦ T1 is also an isomorphism.

13. Let T : P2 → P2 be defined by T (c0 + c1 x + c2 x2 ) = c1 x + 2c2 x2 . Find the matrix [T ]B ,B relative to the bases B = {1, x, x2 } and B = {1 + x, 1 − x2 , x − x2 }.

14. Let T1 : U → V and T2 : V → U be linear transformations, and let B and B be bases for U and V respectively. Compute [T2 ◦ T1 ]B where ⎡ ⎤

1 0 ⎢ ⎥ 1 2 1 ⎥ [T1 ]B ,B = ⎢ . ⎣ 1 2⎦ , [T2 ]B,B = −2 2 1 −1 0

15. Let T : R2 → R2 be the linear operator defined by T (x, y) = 2x − y, 2x + y , and let

1 1 B= , be a basis for R2 .

1 2 (a) Find [T ]B . (b) Use the result in part (a) to compute T (5, −3).

16. Let T : R2 → R3 be a linear transformation,

and let B = {v 1 , v2 } and B = {w1 , w2 , w3 }, 2 3 be bases for R and R , where v1 = 1, 2 , v2 = 1, 3 , w1 = 1, −1, 1 , w2 = 1, 0, 1 , and w3 = 0, 1, −1 . Find a formula for T given that ⎡ ⎤ 2 1 ⎢ ⎥ ⎥.

[T ]B ,B = ⎢ 1 3 ⎣ ⎦ 3 0

17. Let T : V → V be a linear operator, and let B be any basis for V . If T is the identity operator on V and V is n-dimensional, prove that [T ]B is always the n × n identity matrix.

18. Let T : R2 → R2 be the orthogonal projection onto the line y = x, and let B = {u1 , u2 }

1 1 1 0 , u2 = , v1 = , v2 = . and B = {v1 , v2 } be bases for R2 where u1 = 1 −1 0 1 (a) Find [T ]B . (b) Find [T ]B using the equation [T ]B = P −1 [T ]B P .

+ 4x 3x x 1 1 2 = .

19. Let T : R2 → R2 be defined by T x2 4x1 − 3x2 Chapter 8

Elementary Linear Algebra 12e

–5–

Anton/Rorres

(a) Find the eigenvalues of T . (b) Find a basis for each eigenspace of T .

20. Prove that any square matrix is similar to itself.

21. Prove that if A and B are similar matrices, then AT is similar to B T .

22. Let T : V → V be a linear operator and let B and B be bases for V . Let A = [T ]B , P = PB →B , P −1 = PB→B , and C = P −1 AP . (a) Prove that det (A) = det (C). (b) Prove that A and C have the same characteristic polynomial.

23. Find the shear in the y-direction of the vector −3, 1 by the factor k = 2 .

24. Find the standard matrix for the operator that rotates each vector clockwise about the positive y-axis through an angle θ in R3 .

25. If the standard matrix A creates a shear in the x-direction by a factor k, what is the effect of AT ?

26. Find the standard matrix in R3 for the composition of a counterclockwise rotation of 60◦ about the positive y-axis, followed by a projection onto the yz-plane, followed by a reflection across the xy-plane.

27. Let T be defined by the equations x1 + 2x2 + 2x3 = w1 2x1 + x2 + 2x3 = w2 2x1 + 2x2 + 3x3 = w3

Find the inverse operator T −1 w1 , w2 , w3 .

28. Prove that for any angle θ, the standard transformation T : R3 → R3 defined by counterclockwise rotation about the positive x-axis through an angle θ is invertible. √ √ 2 2 − 2 √ on the unit square.

29. Describe the effect of multiplication by the matrix √22 2 2

30. Express A =

−1

1

2

as a product of elementary matrices, and describe the effect of 0 12 multiplication by A in terms of shears, compressions, expansions and reflections.

Chapter 8

Elementary Linear Algebra 12e

–6–

Anton/Rorres

Answers Multiple Choice Answers 1. (B) 2. (D) 3. (A) 4. (D) 5. (B) 6. (A) 7. (C) 8. (C) 9. (C) 10. (B) 11. (D) Free Response Answers 1. Kernel consists of all matrices where a = d.

2. (a) B = 1, 0 , 0, 1 (b) ker(T ) =

0, 0

(c) rank(T ) = 2, nullity(T ) = 0 5. ker(T ) = 0, T is one-to-one 6. T −1 (p(x)) = p(x + 3) 9. All real k = 0, 2, −2.

10. Possible solution: T (a0 + a1 x + a2 x2 + a3 x3 ) =

Chapter 8

a0

a1

a2

a3

Elementary Linear Algebra 12e ⎡ ⎢ 13. [T ]B ,B = ⎢ ⎣0 0

14. [T2 ◦ T1 ]B =

15. (a)

⎤

1 2 − 12 1 2

0

−1

−4

2

4

–7– 1

⎥ −1⎥ ⎦ −1

2

4

−1

4

(b) 13, 7

16. T (x, y) = x + y, 5x − 2y, −8x + 4y

18. (a) [T ]B =

1

0

0

0

(b) [T ]B =

1 2 1 2

1 2 1 2

19. (a) λ1 = 5, λ2 = −5

2 −1 (b) , 1 2

23. −3, −5 ⎡ ⎢ 24. ⎢ ⎣

−sin (θ)

⎤

cos (θ)

0

0

1

⎥ 0⎥ ⎦

sin (θ)

0

cos (θ)

25. Shear in the y-direction by factor k. ⎤

⎡ 0

⎢ 26. ⎢ ⎣ √0 3 2

0 1 0

0

⎥ 0⎥ ⎦

− 12

Chapter 8

Anton/Rorres

Elementary Linear Algebra 12e 27.

–8–

w1 + 2w2 − 2w3 , 2w1 + w2 − 2w3 , −2w1 − 2w2 + 3w3

Anton/Rorres

29. Counterclockwise rotation about the origin through an angle of 45◦ .

30. A =

1

2

−1

0

1

0 1 2

; Compression in the y-direction by a factor of 12 , then

0 1 0 1 0 reflection about the y-axis, then shear in the positive x-direction by a factor of 2.

Chapter 8

Chapter 9: Numerical Methods Multiple Choice Questions 1. Which of the following is the matrix P in the P LU -decomposition of the following matrix? ⎡ ⎤ 0 2 4 ⎢ ⎥ ⎢ 2 4 3⎥ ⎣ ⎦ 1 2 3 ⎤ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎡ 1 0 0 0 0 1 0 0 1 0 1 0 ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ (B) ⎢0 1 0⎥ (C) ⎢1 0 0⎥ (D) ⎢0 1 0⎥ (A) ⎢ 0 0 1 ⎦ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎣ 0 0 1 0 1 0 1 0 0 1 0 0 2. Let A be a square matrix with P LU -decomposition A = P LU . Suppose Ax = b. Which of the following expressions is not equivalent to P −1 b?

(A) Ax (B) LU x (C) P −1 Ax (D) All of the above are equivalent to P −1 b.

3. The distinct eigenvalues of a matrix are λ1 = 4, λ2 = 0, and λ3 = −5. Which eigenvalue is the dominant eigenvalue?

(A) λ1 (B) λ2 (C) λ3 (D) There is no dominant eigenvalue.

4. Which of the following is a dominant eigenvector of the following matrix? ⎡ ⎤ 3 0 0 1 ⎢ ⎥ ⎢0 −1 2 0⎥ ⎢ ⎥ ⎢ ⎥ ⎢0 ⎥ 0 −2 −1 ⎣ ⎦ 0 0 0 5 ⎡ ⎤ ⎡ ⎤ ⎤ ⎡ ⎤ ⎡ 0 0 1 −21 ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ ⎢ 1⎥ ⎢ 2⎥ ⎢ 0⎥ ⎢ 2⎥ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ (A) ⎢ ⎥ (B) ⎢ ⎥ (C) ⎢ ⎥ (D) ⎢ ⎥ ⎢ 0⎥ ⎢−1⎥ ⎢ 0⎥ ⎢ 6⎥ ⎣ ⎦ ⎣ ⎦ ⎦ ⎣ ⎦ ⎣ 0 0 0 −42

Elementary Linear Algebra 12e

–2–

Anton/Rorres

5. Let A and B be n × n matrices, and let c be a real scalar. Which of the following operations requires the most flops to compute?

(A) AB (B) cA (C) A + B (D) tr (A2 ) 6. Let A be an n × n matrix. What is the approximate cost (in flops) of computing an LU -decomposition of A? (A) ≈ n2 (B) ≈ 2n3 (C) ≈ 23 n3 (D) ≈ 2n2 7. What are the singular values of the following matrix? ⎤ ⎡ 1 −1 ⎥ ⎢ ⎢ 1 −1⎥ ⎦ ⎣√ 3 0 √ √ (A) 1 and 6 (B) 0, 1, and 6 (C) 1 and 6 (D) 0, 1, and 6 8. Let A be an m × n matrix. Which of the following statements is not always true?

(A) A and AT A have the same null space.

(B) A and AT A have the same column space.

(C) A and AT A have the same row space.

(D) A and AT A have the same rank.

9. Let A be an m × n matrix and let its reduced singular value decomposition be given by A = U1 Σ1 V1T . If A has k singular values, what is the dimension of the matrix V1 ?

(A) k × n (B) n × k (C) k × k (D) n × n

Chapter 9

Elementary Linear Algebra 12e

–3–

Anton/Rorres

Free Response Questions 1. Find an LU -decomposition of the following matrix. ⎤ ⎡ 1 1 1 −1 ⎥ ⎢ ⎢ 2 5 2 −2⎥ ⎥ ⎢ ⎥ ⎢ ⎢ 0 1 2 −4⎥ ⎦ ⎣ −3 1 0 0 2. Use an LU -decomposition to solve the following system. ⎡ ⎤⎡ ⎤ ⎡ ⎤ 2 1 2 −1 x1 ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢3 7 −1⎥ ⎢x2 ⎥ = ⎢ 14⎥ ⎣ ⎦⎣ ⎦ ⎣ ⎦ −4 2 6 −6 x3 3. Suppose a matrix A has the following LU -decomposition: ⎤⎡ ⎡ ⎤ 1 −3 2 1 0 0 ⎥⎢ ⎢ ⎥ 1⎥ ⎥ ⎢0 A=⎢ 1 − 2 6 0 ⎦⎣ ⎣ 2⎦ 3 2 2 0 0 1 (a) Compute L−1 . (b) Compute U −1 . (c) Use the result in parts (a) and (b) to find A−1 .

4. Let A be an n×n matrix such that it has an LDU decomposition A = LDU . If D = [dij ], prove that det (A) = d11 d22 · · · dnn .

5. Use the power method with Euclidean scaling starting at x0 and stopping at x4 to approximate a dominant eigenvector and the dominant eigenvalue of the following matrix.

1 1 4 ; x0 = 0 2 3 6. Use the power method with maximum entry scaling starting at x0 and stopping at x4 to approximate a dominant eigenvector and the dominant eigenvalue of the following matrix. ⎡ ⎤ ⎡ ⎤ 1 0 11 −5 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢−2 17 −7⎥ ⎣ ⎦ ; x 0 = ⎣ 1⎦ −4 26 −10 1

Chapter 9

Elementary Linear Algebra 12e

7. Let A =

2

0

0

−2

–4–

Anton/Rorres

. Explain why the power method may fail to converge if applied to A.

8. Let A = kI where I is the n × n identity matrix and k is a positive real scalar. (a) Show that if the power method with Euclidean scaling is applied to A with initial An x 0 guess x0 , then xn = = x0 for any positive integer n. ||An x0 || (b) Show that if the power method with maximum entry scaling is applied to A with An x 0 initial guess x0 , then xn = = x0 for any positive integer n. max(An x0 ) (c) Hence, show that the power method converges to an eigenvector v of A if and only if x0 = v.

9. If A is an 100 × 100 matrix, approximately how many flops are required to solve Ax = b by computing the value of x = A−1 b?

10. Let A and B be n×m matrices, and let c be a real scalar. What is the minimum number of flops required to compute: (a) c(A + B)? (b) cA + cB? 11. Let Ax = b be a system of n equations in n unknowns. The number of flops required to solve the system using Cramer’s rule is 1 (4n4 − n3 + 2n2 + 5n − 6).

6 If a certain computer can process 70 teraflops per second, how many hours will the computer take to process a system where n = 3 · 104 ?

12. If A is a 2 × 3 matrix and AT A has eigenvalues λ = 2, 0, and 3, what are the singular values of A? 13. Let A be a 3 × 2 matrix. If the eigenvalues of AT A are λ1 = 4 and λ2 = 1. What is Σ in the singular value decomposition of A = U ΣV T ?

14. Find the singular values of the following matrix.

3 1 1 −1

3

1

15. Find a singular value decomposition of the following matrix.

3 2 2 2

3

−2

Chapter 9

Elementary Linear Algebra 12e

–5–

Anton/Rorres

16. Consider the following matrix. ⎡ 1 ⎢ ⎢ 1 ⎢ A=⎢ ⎢−1 ⎣ 0

⎤ 0

⎥ 1⎥ ⎥ ⎥ 1⎥ ⎦ 1

(a) Find the reduced singular value decomposition of A. (b) Find the reduced singular value expansion of A.

Chapter 9

Elementary Linear Algebra 12e

–6–

Anton/Rorres

Answers Multiple Choice Answers 1. (A) 2. (A) 3. (C) 4. (A) 5. (A) 6. (C) 7. (C) 8. (B) 9. (B) Free Response Answers

⎤

⎡

1 ⎢ ⎢ 2 ⎢ 1. Possible answer: L = ⎢ ⎢ 0 ⎣ −3

0

0

3

0

1

2

4

3

⎡

0

1 ⎥ ⎢ ⎥ ⎢ 0⎥ ⎢0 ⎥, U = ⎢ ⎢0 0⎥ ⎦ ⎣ 3 0

⎡ ⎤ ⎡ ⎤ x1 1 ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ 2. ⎢ ⎣ x2 ⎦ = ⎣ 2⎦ 3 x3 ⎡

1

0

0

− 76

1 6 − 16

1 2

⎢ 1 3. (a) ⎢ ⎣− 3 ⎡

1

3

⎤

⎥ 0⎥ ⎦

− 12

⎤

⎢ (b) ⎢ ⎣0

1

1⎥ 2⎦

0

0

1

⎥

Chapter 9

1

1

1

0

0

1

0

0

−1

⎤

⎥ 0⎥ ⎥ ⎥ −2⎥ ⎦ 1

Elementary Linear Algebra 12e ⎡

7 ⎢ 12 11 (c) ⎢ ⎣− 12 − 76

5. v ≈

7 12 1 12 − 16

0.7088

− 14

–7–

Anton/Rorres

⎤ ⎥

1⎥ 4⎦ 1 2

, λ ≈ 4.9952

0.7054 ⎤

⎡ 0.4079

⎥ ⎢ ⎥ , λ ≈ 4.0965 6. v ≈ ⎢ 0.6053 ⎦ ⎣ 1 9. ≈ 2, 000, 000 flops 10. (a) 2nm flops (b) 3nm flops 11. 2.143 hours √ √ 12. σ1 = 2, σ2 = 0, σ3 = 3 ⎡

2 ⎢ ⎢ 13. Σ = ⎣0 0

0

⎤

⎥ 1⎥ ⎦ 0

14. σ1 = 0, σ2 =

15. U ΣV T =

√

√1 2 1 √ 2

10, σ3 = − √12 √1 2

√

12 ⎡

5

0

0

0

3

0

√1 ⎢ 2 ⎢ √1 ⎣ 2

0

− 3√1 2 1 √ 3 2 − 3√4 2

0

0

1

3

1

0

⎤

⎡ 0

⎢ ⎢ √1 ⎢ 16. (a) U1 Σ1 V1T = ⎢ 13 ⎢√ ⎣ 3 √1 3

√1 3⎥ √ √1 ⎥ 3 ⎥ 3 ⎥ 0 − √13 ⎥ ⎦

√

0

Chapter 9

⎤T

− 23 ⎥ 2⎥ 3⎦ 1 3

Elementary Linear Algebra 12e ⎡

⎤

0

⎢ √ ⎢ ⎢0 (b) A = 3 ⎢ ⎢0 ⎣ 0

–8– ⎡

√1 ⎢ 3 ⎥ 1 √1 ⎥ √ ⎢ ⎢ √3 3⎥ 3 + ⎢ ⎥ ⎢− √1 √1 ⎥ ⎣ 3 3⎦ √1 0 3

0

⎤ 0

⎥ 0⎥ ⎥ ⎥ 0⎥ ⎦ 0

Chapter 9

Anton/Rorres