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) Suppose dim(V ) = dim(W) = n. Let T : V → W be linear. Prove that the following statements are equivalent: a) T : V → W is invertible. b) For every basis β of V , T(β) is a basis of W. c) There exists a basis β of V such that T(β) is a basis of W. Hint: Prove a) ⇒ b) ⇒ c) ⇒ a). 2) Let V and W be finite-dimensional vector spaces. Let dim(V ) = n, dim(W) = m. Let β = (v1, · · · , vn) and γ = (w1, · · · , wm) be ordered bases for V and W, respectively. Let [[] γ β : L(V, W) → Mm×n(F) be the function which sends T to [T] γ β. Prove that [[] γ β is a linear transformation. 3) Let A ∈ Mn×n(F). Let γ = (e1, · · · , en) be the standard ordered basis, and let β = (v1, · · · , vn) be any ordered basis. Let v ∈ Fn . Prove the following: i) [v]γ = v. ii) [A] γ γ = A. iii) [In] β β = In. iv) [0] β β = 0. v) [In] γ β is invertible with inverse [In] β γ.1 4) Let A ∈ Mn×n(F). Prove that A is invertible if and only if the columns of A form a basis for Fn. 5) Let V and W be finite-dimensional with ordered bases β and γ, respectively. Let T : V → W be a linear transformation. Suppose that rk(T) = k. Prove that [T] γ β has at least k nonzero entries. 6a) Define the trace of a matrix A ∈ Mn×n(F) to be sum of the diagonal entries of A: tr(A) := Pni=1 Aii. If A, B ∈ Mn×n(F), prove that tr(AB) = tr(BA). b) Prove that if A is similar to B, then tr(A) = tr(B). 7) Suppose V is finite-dimensional. Let T : V → V be linear. Suppose that rk(T) = rk(T2). Prove that ker(T) ∩ im(T) = {0}. Hint: Let T0 be the restriction of T to im(T). Prove that: i) im(T0 ) = im(T2) ii) ker(T0 ) = ker(T) ∩ im(T) iii) Apply the Rank-Nullity Theorem to T0. 8) Let V and W be finite-dimensional with ordered bases β = (v1, · · · , vn) and γ = (w1, · · · , wm). Let T : V → W be linear. Prove that rk(T) = rk( [T] γ β). Computational Problems You don’t need to prove your answers to the following questions, but you should still show your work. 9a) Find a basis for the image and kernel of A =  1 0 2 2 1 3 鮶1 0 02 4 3 5 1 1 1 . b) Find the set of all solutions x to the equation Ax = b, where b = 10000. 2 c) Find the set of all solutions x to the equation Ax = b, where b =  11鮶110 . 10) Define T : M2×2(R) → P2(R) by T a b c d = (a + b) + (2d)x + bx2 Let β = 1 0 0 0 ,  0 1 0 0 ,  0 0 1 0 ,  0 0 0 1, γ = (1, x, x2). Compute [T] γ β. 11) Let λ ∈ F. Let T : V → V be a linear transformation. A nonzero vector v ∈ V is a λ-eigenvector of T if T(v) = λv. Let A ∈ M3×3(R). Let v1 = 101 , v2 = 솬213  , v3 = 鯦410 . Then β = (v1, v2, v3) is a basis for R3 . Let γ = (e1, e2, e3) be the standard ordered basis of R3 . Suppose v1 is a 2-eigenvector of A, v2 is a 3-eigenvector of A, and v3 is a 鯦1-eigenvector of A. a) Find [A] β β. b) Find [I3] γ β c) Find [I3] β γ d) Find A. 12) Let β = 1 0 0 0 ,  0 1 0 0 ,  0 0 1 0 ,  0 0 0 1. Let T : M2×2(R) → M2×2(R) be the linear transformation given by T(A) = At . Find [T] β β.