Linear Algebra homework

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Math 235 - Summer 2015 Homework 2

Due Monday June 8 in class

Remember: In this course, you must always show reasoning for your answers. You can use any result we have proved in class, in textbook reading, or in a previous homework.

Problem 1 For each of the following problems, you must justify your answer by finding the general solution to the corresponding system of linear equations, or by showing that no solution exists.

(a) In the vector space P3(R), can −2x3 − 11x2 + 3x+ 2 be written as a linear combination of vectors in {x3 − 2x2 + 3x− 1, 2x3 + x2 + 3x− 2}?

(b) In the vector space M2×2(R), can (

1 0 0 1

) be written as a linear combination of vectors

{( 1 0 −1 0

) ,

( 0 1 0 1

) ,

( 1 1 0 0

)} ?

Problem 2 Show that a subset W of a vector space V (over a field F ) is a subspace of V if and only if span(W ) = W .

Problem 3 You are given a subset S of a vector space V . Determine whether S is linearly dependent or linearly independent using exclusively methods developed in this course, and justify your answers.

(a) V = R3 and S = {(1, 2,−1), (2,−3, 1), (2, 3,−5)}. (b) V = P3(R) and S = {1, 1 + 2t+ t2, 1− 2t+ t3, t2 + t3}. (c) V = F(R,R) and S = {t, et, sin(t)}.

Problem 4 Prove that a subset S of a vector space V is linearly dependent if and only if there exists a proper subset S′ ( S with the same span as S.

Problem 5 Exercise 1.6.13 from the textbook.

Problem 6 You are given a subspace S of M2×2(F ), the vector space of 2 × 2 matrices with entries in a field F . You are required to find a basis for this subspace, and to find the dimension of this subspace. For each problem, you DO NOT need to prove that S is a subspace, but you DO need to prove that your conjectured basis is, in fact, a basis (that is, you need to show it is a linearly independent generating set for S).

(a) S is the subspace of all diagonal 2× 2 matrices with entries in F . (b) S is the subspace of all symmetric 2× 2 matrices with entries in F . (c) S is the subspace of all skew-symmetric 2× 2 matrices with entries in F .

Problem 7 Let W1 and W2 be subspaces of a finite-dimensional vector space V . Prove that dim(W1∩W2) ≤ min{dim(W1),dim(W2)} and dim(W1 +W2) ≥ max{dim(W1),dim(W2)}.

Problem 8 Each of the maps below goes from one vector space to another (where both vectors spaces are over the same field). For each map: prove that it is linear, determine whether it is one-to-one or not (prove your answer), and determine whether it is onto or not (prove your answer).

(a) T : P3(R)→M2×2(R) defined by T (p) = ( p(0) p′(0) p′′(0) p′′′(0)

) .

(b) T : M2×2(F ) → F defined by T (A) = tr(A), where F is a field. (Recall that for an n × n matrix, tr(A) =

∑n i=1Aii.)

Problem 9 Suppose that T : R2 → R2 is linear and that T ((1, 2)) = (3, 4) and T ((1, 3)) = (0, 1). Find T ((1, 0)). Is T one-to-one? Justify your answer.

Problem 10 Let V and W be finite-dimensional vector spaces over the same field F and let T : V →W be a one-to-one linear transformation.

(a) Let L ⊆ V . Prove that L is linearly independent if and only if T (L) ⊆W is linearly independent. (b) Suppose that dim(V ) = dim(W ) and assume β ⊆ V . Prove that β is a basis for V if and only if T (β)

is a basis for W . (c) Deduce that if dim(V ) = dim(W ), then there exist ordered bases β and γ for V and W respectively

such that [T ]γβ is the identity matrix. (Hint: Let β be any ordered basis of V . One by one, choose the

elements of γ so that it is a basis for W and so that the ith column of [T ]γβ is ei.)

Problem 11 In each of the following parts you are given vector spaces V and W , the respective standard ordered bases β and γ for these vector spaces, and a linear transformation T : V →W . Determine [T ]γβ , that is, the matrix representation of the linear transformation T in the ordered bases β and γ.

(a) V = R3, β = {e1, e2, e3}, W = P2(R), γ = {1, t, t2}, and T ((a, b, c)) = a + ct2. (Recall ei ∈ Rn is the n-tuple whose ith entry is a 1 and all other entries are 0s.)

(b) V = R2, W = R, β = {e1, e2}, γ = {e1}, and letting λ ∈ R and b = (b1, b2) ∈ R2, T is defined by T (v) = λ(v · b), where · is the dot product in R2.

a b c d

)) = (a + d, b +

c, a+ b+ c+ d). (Recall Eij is the matrix whose (i, j) entry is a 1 and all other entries are 0s.)

Problem 12 In each of the following parts you are given vector spaces V and W , respective ordered bases β and γ for these vector spaces, and a linear transformation T : V → W . Determine the matrix [T ]γβ of the linear transformation with respect to the ordered bases β and γ.

(a) V = R3, β = {e1 + e2, e2, e3 − e1}, W = R2, γ = {e1, 2e2}, and T ((a, b, c)) = (a, b+ c).

(b) V = M2×2(R), β = {2E11, E21−E22, E12, E22}, W = R3, γ = {e3, e2, e1}, and T ((

a b c d

)) = (a, b, c).