Linear Algebra Assignment Help

Linear Algebra Assignment Help

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Mathematics With Applications Linear Algebra And The Calculus Of Variations Problem 1: Consider the set V of all 2 × 2 skew symmetric matrices ⎠

with addition and multiplication by a scalar defined in the “natural way”. a)Show that V is a vector space. b)Give an example of a set of 2 linearly dependent vectors in V, and an example of 2 linearly independent vectors in V. c)What is the dimension of V?

⎛ d)If x and y are two vectors in V, show that ⎝

⎞ ⎠

x · y = x 12 y12 + x 21 y21 is a proper definition of a scalar product on V. From hereon assume that the vector space V has been made Euclidean with this scalar product. e)Find an orthonormal basis for V. f) Let A be the transformation defined by

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Show that A is a tensor. g)Show that the tensor A is symmetric. h)Is A singular or nonsingular? i) Calculate the eigenvalues of A .

Problem 2: Consider the 3-dimensional Euclidean vector space V which is comprised of all polyno mials of degree ≤ 2; a typical vector in V has the form x = x(t) = co + c1t + c2 t2 .

Addition and multiplication by a scalar are defined in the “natural way”. The scalar product between two vectors x and y is defined as

Problem 3: Let A be a symmetric positive definite tensor on a n-dimensional vector space V. Let α 1 , α 2 , . . . , α n be the eigenvalues of A where the eigenvalues are ordered according to 0 < α1 ≤ α2 . . . ≤ α n .

Show that the smallest eigenvalue

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where the minimization is taken over all unit vectors x ∈ V. (Hint: Work using the components of A and x in a principal basis of A.) Show similarly that the largest eigenvalue

where the maximization is taken over all unit vectors x ∈ V. Problem 4: If A is an arbitrary nonsingular tensor, show that

.

Problem 5: Give an example of a tensor A (that is NOT the identity tensor I ) which has the property A = A2 = A3 = A4 = . . .

Hint: Think geometrically. a unit vector x such that (c · x)(d · x) > 0; and that one can always find some other unit vector x Problem 6: Let c and d be two distinct non-zero vectors belonging to a 3-dimensional Euclidean such that (c ·Show x )(d · x ) < 0. vector space. (geometrically by drawing arrows or some other way) that one can always find Let C be the symmetric tensor defined by C = I + c ⊗ d + d ⊗ c.

Show (using the result of Problem 3 or otherwise) that the eigenvalues λ 1 , λ 2 , λ 3 of C have the property that λ 1 < 0, λ 2 = 0, λ 3 > 0. mathsassignmenthelp.com

Solution Problem 1: Consider the set V of all 2 × 2 skew-symmetric matrices, i.e. matrices of the form

⎝ a) Define the addition of two vectors by ⎛ ⎝

⎞ ⎠

⎠ , ⎛ ⎝

⎞ ⎠

⎛ ⎝

⎞ ⎠

and the multiplication of a vector x by the scalar α as ⎞ ⎠ .

⎠ Then one can readily verify that x + y ∈ V whenever x and y ∈ V, and that αx ∈ V for every scalar α whenever x ∈ V . Thus V is a vector space. b) Since

it follows that yx + (−x)y = o for any two vectors x and y. Thus every pair of vectors is for all x , y, z ∈ V and all scalars α, β . Thus this is a proper definition of a scalar product.

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linearly dependent. There is no pair of vectors that is linearly independent. c)It follows from item (b) that the dimension of the vector space is 1. d)If x and y are two vectors in V, and we define

then we can verify that x · y has the properties a) x · y = y · x , b) x · (αy + βz ) = αx · y + βx · z , c) x · x ≥ 0 with x · x = 0 if and only if x = o, e) Since the dimension of V is 1, any one (non null) vector provides a basis for it. Pick

Note that the length of eJ is

Thus a basis composed of a unit vector is {e} where

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f ) Let A be the transformation defined by ⎛ ⎝

⎞ ⎠

We can readily verify that (i) the vector A x ∈ V for every x ∈ V, and (ii) that A(αx + βy) = α A x + βAy for all vectors x, y and all scalars α, β. ⎛Therefore A is a linear transformation (tensor). . ⎝

g)From item (d), a · b = 2(a)(b) for any two vectors a and b. Thus it follows that ⎛ A x · y ⎞= 2(2x)(y) = 4xy ⎠ ⎝ while

x · Ay = 2(x)(2y) = 4xy.

Thus A x · y = x · Ay and so A is symmetric. h)If A x = o then 2x = 0 and so x = 0 from which it follows that x = o. Thus A x = o if and only if x = o. Thus A is non-singular. i) To find the eigenvalues of A we must find λ such that A x = λx, i.e. ⎝

⎠.

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Problem 2: Consider the 3-dimensional Euclidean vector space V which is comprised of all polyno mials of degree ≤ 2; a typical vector in V has the form Addition and multiplication by a scalar are defined in the “natural way”. between two vectors x and y is defined as

Clearly, f 1 = 1,

f 2 = t,f 3 = t2 ,

(2.3)

is a basis for V (since, as one can readily verify, they form a set of three linearly independent vectors). We now consider the following three vectors,

and choose α, β and γ such that the three e’s are orthogonal to each other.

Note that

First, enforcing e2 · e1 = 0 leads to

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whence α = 0 and therefore

Next enforcing e3 · e1 = 0 leads to whence γ = −1/3. Finally enforcing e3 · e2 = 0 leads to whence β = 0. Thus The triplet of vectors e1, e2, e3 are mutually orthogonal. In order to generate an orthonormal basis, we have to make each of these vectors a unit vector. To this end we calculate

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Problem 3: Since A is symmetric it has n real eigenvalues α1 , α 2 , . . . , α n and a corresponding set of orthonormal eigenvectors {a 1 , a 2 , . . . a n }. The eigenvalues can be ordered as α1 ≤ α2 . . . ≤ α n .

Throughout this problem we use {a 1 , a 2 , . . . a n } as the basis with respect to which we calculate all components of vectors and tensors. In particular, we know that the matrix of components of A in this basis is diagonal and that A i i = α i . Let x i be the ith component of a unit vector x. Then

which in this principal basis of A simplifies to

Therefore for all unit vectors x. Since α1 = x · Ax when x = a1 it follows that

where the minimization is taken over all unit vectors x ∈ V. Similarly, since α1 ≤ α2 . . . ≤ αn it follows that mathsassignmenthelp.com

for all unit vectors x. Since αn = x · Ax when x = an it follows that

where the maximization is taken over all unit vectors x ∈ V Problem 4: For any two vectors a and b, by the definition of the transpose of a tensor A we have A a · b = a · A T b.

Let

(4.1)

Since A (and therefore AT ) are non-singular (4.2) gives

Substituting (4.3) into (4.1) gives Next, applying the identity (4.1) to the tensor A−1 allows us to write the last term in (4.4) as mathsassignmenthelp.com

Combining, (4.4) and (4.5) with gives

Since this must hold for all vectors x and y it follows that

Problem 5: Motivated by geometry, consider the projection tensor

where n is a unit vector. Then

Operating on this by Π gives Π2 = Π3 and operating on that by Π gives Π3 = Π4 and so on. Thus, for the projection tensor Π

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Problem 6: Since c and d are distinct non-zero vectors belonging to a 3-dimensional vector space, they define a plane in that space. The figure shows this plane; the figure on the left corresponds to the case when the angle between these two vectors is obtuse while the figure on the right corresponds to the case when the angle is acute. The unit vectors x 1 , x 2 , x 3 , x 4 have been selected such that the scalar products c · x k and d · x k have definite signs, either positive or negative as shown in the figure. In the case depicted on the left, (c · x 3 )(d · x 3 ) > 0 while (c · x 4 )(d · x 4 ) < 0. Similarly in the case depicted on the right, (c · x 1 )(d · x 1 ) > 0 while (c · x 2 )(d · x 2 ) < 0. Thus in either case, one can always find a unit vector x that makes (c · x)(d · x) > 0, and some other unit vector x

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Figure 6.1: The plane defined by the vectors c and d with four unit vectors x1, x2, x3, x4 for which the scalar products c · xk and d · xk have definite signs, either positive or negative.

Let B be the symmetric tensor defined by Since B is symmetric it has three real eigenvalues which we can order as β1 ≤ β2 ≤ β3. First: for any unit vector x From Problem 3 we know that where the minimization is taken over all unit vectors x ∈ V. However from the first part of this problem we know that there is some unit vector x for which (d·x)(c·x) < 0. Therefore (d·x)(c·x) is negative for some value of x and so its minimum value is necessarily negative as well. Thus In an entirely analogous way, from Problem 3 we know that

where the maximization is taken over all unit vectors x ∈ V. However from the first part of this problem we know that there is some unit vector x for which (d·x) (c·x) > 0. Therefore (d·x)(c·x) is positive for some value of x and so its maximum value is necessarily positive as well. Thus mathsassignmenthelp.com

Finally, as noted previously the vectors c and d define a plane. Let e be a vector that is normal to that plane so that e · c = e · d = 0. Then B e = (d · e)c + (c · e)d = o

(6.7)

(which we can write as B e = 0e) and so 0 is an eigenvalue of B . Thus β2 = 0.

(6.8)

The eigenvalues λ i of the symmetric tensor C defined by C = I + c⊗d+ d⊗c= I + B are related to the eigenvalues β i of B by λ i = 1 + β i . Thus the eigenvalues of C have the property that λ 1 < 1, λ 2 = 1, λ 3 > 1.

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(6.9)