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. 1a) Let W be a subspace of V . Prove that the annihilator W◦ ⊆ V ∗ is a sub￾space. b) Let T : V1 → V2 be a linear transformation. Prove that the dual T∗ : V ∗2 → V ∗1 is linear. c) Prove that for each v ∈ V , the evaluation map evv : V ∗ → F is linear, hence evv ∈ V ∗∗ . d) Prove that the map ev : V → V ∗∗ which sends v to evv is linear. 2) Let V be a finite-dimensional vector space. Let β = (v1, · · · , vn) be an or￾dered basis for V . Prove that the dual basis β∗ = (f1, · · · , fn) is a basis for V ∗. 3) Let V, W be a finite-dimensional vector spaces. Let T : V → W be a lin￾ear transformation. Prove that: i) ker(T∗ ) = im(T)◦ ii) n(T∗ ) = n(T) + dim(W) dim(V ) iii) rk(T∗ ) = rk(T) iv) im(T∗ ) = ker(T)◦ (Hint: Show one inclusion and make a dimension argu￾ment!) 4) Let β = (v1, · · · , vn) be an ordered basis for V and let β∗ = (f1, · · · , fn) 1 be the dual basis for V ∗ . Let g ∈ V ∗ . Prove that g = Pni=1 g(vi)fi. 5) Let V be finite-dimensional and let W ⊆ V be a subspace. Let v ∈ V and suppose that f(v) = 0 for all f ∈ W◦ . Prove that v ∈ W. 6) Let V be finite-dimensional, and let W1, W2 ⊆ V be subspaces. Prove that W1 ⊆ W2 if and only if W◦2 ⊆ W◦1 . 7) Let W ⊆ V be a subspace. Let v1, v2 ∈ V . Prove that the following statements are equivalent: i) v1 + W = v2 + W ii) v1 v2 ∈ W iii) (v1 + W) ∩ (v2 + W) = ∅ Computational Problems You don’t need to prove your answers to the following questions, but you should still show your work. 8) Let γ = (e1, e2, e3, e4) be the standard basis of R4 . Let β = {1} be the standard basis of F. Calculate [f] β γ, where f ∈ (R4)∗ is the functional given by f(a, b, c, d) = 3a 2b + 6c d. 9) Let γ = (e1, e2, e3) be the standard basis of R3 . Define a functional f ∈ (R3)∗ by setting f(e1) = f(e2) = f(e3) = 1. Compute ker(f). Does there exist an injective functional g ∈ (R3)∗?  