Proof Problems Your answers to the problems in this section should be proofs, unless otherwise stated. F is a field, V and W are vector spaces over F. 1) Proposition 14.4 of the Lecture Notes is false when V is an infinite dimensional vector space! Consider the following example: Let V = {(a1, a2, · · · , an, · · ·) ∈ F∞ | ai = 0 for all but finitely many i} be the vector space of eventually-zero sequences. Let ei ∈ V be the sequence with a 1 in position i and 0 everywhere else. Define fi : V → F by setting fi(ej ) = δij . Thus γ = (f1, f2, · · · , fn, · · ·) is the “dual basis” of γ. a) Prove that γ = (e1, e2, · · · , en, · · ·) is an ordered basis for V . b) Prove that γ is linearly independent in V but does not span V . (In other words, γ is not actually a basis for V !) 2) Let V and W be finite-dimensional vector spaces. Let T : V → W be a linear transformation. a) Prove that T is surjective if and only if T is injective. b) Prove that T is injective if and only if T is surjective. 3) Let T : V → W be a linear transformation. Recall the evaluation maps ev : V → V and ev : W → W defined in Theorem 14.11 of the Lecture Notes. Prove that the following diagram commutes: 1 V W V W T ev ev T That is, prove that ev T = T ev. 4) Let V be a finite-dimensional vector space and let W V be a subspace. Consider (W ) V , i.e. (W ) = {α ∈ V | f ∈ W , α(f) = 0}. Let ev : V → V be the evaluation map. Prove that ev(W) = (W ) . 5) Let T : V → W be a linear transformation. Prove that the map T : V/ker(T) ～ → im(T) given by T(v + ker(T)) = T(v) is well-defined and an isomorphism. 6) Let T : V → V be a linear operator. Let W V be a subspace. We say W is T-invariant if T(w) ∈ W for all w ∈ W. a) Prove that im(T) and ker(T) are T-invariant. b) Prove that W is T-invariant if and only if the induced map T : V/W → V/W given by T(v + W) = T(v) + W is well-defined. 7) Let T : V → V be a linear operator and let W V be a T-invariant subspace. Let T |W : W → W denote the restriction of T to W. Prove that i) ker(T |W ) = ker(T) ∩ W ii) im(T |W ) = T(W) 8) Let W be a subspace of a vector space V . Let β = {wi}i∈I be a (possibly infinite) basis of W. Let γ = {vj}j∈J be a disjoint linearly independent set such that β ∪ γ is a basis for V . Let π : V → V/W be the quotient map. a) Prove that π(γ) = {vj + W}j∈J is a basis for V/W. b) Prove that V ～= W × V/W. 9) Let dim(V ) = n, and let T : V → V be a linear operator. Let W V be a T-invariant subspace of dimension k. Let γ1 = (w1, · · · , wk) be an ordered basis for W. Let γ2 = (vk+1, · · · , vn) be a disjoint linearly independent set such that β = (w1, · · · , wk, vk+1, · · · , vn) is an ordered basis of V . Let T |W : W → W denote the restriction of T to W, and let T : V/W → V/W be the induced map defined in Problem 6. Let π : V → V/W; recall from 2 Problem 8 that π(γ2) is an ordered basis of V/W. Write the n × n matrix [T] β β in block form: [T] β β = A B C D Here A is a k×k matrix, B is k×(n k), C is (n k)×k and D is (n k)×(n k). Prove that: i) A = [T|W ] γ1 γ1 ii) C = 0 iii) D = [T] π(γ2) π(γ2) Computational Problems You don’t need to prove your answers to the following questions, but you should still show your work. 10) Let γ = (e1, e2, e3) be the standard basis for R3 . Let γ = (f1, f2, f3) be the dual basis of γ. In this problem we shall consider the following question: What happens to the dual basis if we change only a single vector in γ Let β = (e1, e2, e1 + e3). Clearly β is also a basis for R3 . Let β = (g1, g2, g3) be the dual basis of β. Find a formula for each functional gi and fi . In other words, find an expression for gi a1 a2 a3 in terms of a1, a2, and a3. Note that β differs from γ only in the third vector. Is g1 = f1 and g2 = f2 Explain why or why not.
