A finite dimensional vector space
2. [5 pts] Let V be a finite dimensional vector space and β be an ordered basis for V . Let T : V → V be a linear transformation. Use the principle of mathematical induction to prove [T k ] β β = [T] β β k for all nonnegative integers k.
3. Let β = {1, x, x2} and γ = {E 1,1 , E1,2 , E2,1 , E2,2} be the standard bases for P2(R) and M2×2(R), respectively. Let T : P2(R) → M2×2(R) be defined via T(a0 + a1x + a2x 2 ) = a0 −2a1 −a2 a2 . (a) [2 pts] Let p(x) = 1 − 2x + 4x 2 . Compute [p(x)]β and [T(p(x))]γ. (b) [4 pts] Compute [T] γ β . (c) [2 pts] Compute [T] γ β [p(x)]β using matrix multiplication. Verify that it equals [T(p(x))]γ.
4. [5 pts] Let W be a vector space and let T : W → W be linear. Prove that T 2 = T0 if and only if R(T) ⊆ N(T). (Recall T0 denotes the zero transformation.)
5. Let V , W, and Z be vector spaces, and let T : V → W and U : W → Z be linear. (a) [4 pts] Prove that if UT is one-to-one, then T is one-to-one. (b) [3 pts] If UT is one-to-one, then it is not the case that U must one-to-one. Construct an example of transformations U and T where UT is one-to-one, but U is not one-to-one.
