University Of California A Finite Dimensional Vector Space Questions

2. [5 pts] Let V be a finite dimensional vector space and ? be an ordered basis for V . LetT : V ? V be a linear transformation. Use the principle of mathematical induction toprove [Tk]?? = [T]?? kfor all nonnegative integers k.

3. Let ? = {1, x, x2} and ? = {E1,1, E1,2, E2,1, E2,2} be the standard bases for P2(R) andM2×2(R), respectively. Let T : P2(R) ? M2×2(R) be defined viaT(a0 + a1x + a2x2) = a0 ?2a1?a2 a2 .(a) [2 pts] Let p(x) = 1 ? 2x + 4×2. Compute [p(x)]? and [T(p(x))]?.(b) [4 pts] Compute [T]??.(c) [2 pts] Compute [T]??[p(x)]? using matrix multiplication. Verify that it equals[T(p(x))]?.

4. [5 pts] Let W be a vector space and let T : W ? W be linear. Prove that T2 = T0 ifand 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 isnot one-to-one.