University Of California A Finite Dimensional Vector Space Questions

2. [5 pts] Let V be a finite dimensional vector space and β be an ordered basis for V . LetT : V → V be a linear transformation. Use the principle of mathematical induction toprove [Tk]ββ =[T]ββkfor all nonnegative integers k. 3. Let β = {1, x, x2} and γ = {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)]β and [T(p(x))]γ.(b) [4 pts] Compute [T]γβ.(c) [2 pts] Compute [T]γβ[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 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. 2. [5 pts) Let V be a finite dimensional vector space and 3 be an ordered basis for V. Let
T:V+V be a linear transformation. Use the principle of mathematical induction to
prove [T”)} = (179%)* for all nonnegative integers k.
3. Let B = {1, 2, 22} and y = {E1,1, E1.2, E2,1, E2,2} be the standard bases for P(R) and
M2x2(R), respectively. Let T: P(R) + M2x2(R) be defined via
-2a1
T(ao + aja +aza”) = (
do
-02
“).
012
(a) [2 pts] Let p(x) = 1 – 2x + 4z2. 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()]
4. [5 pts] Let W be a vector space and let T: W + W be linear. Prove that T² = T, if
and only if R(T) = N(T). (Recall T, 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. Con-
struct an example of transformations U and T where UT is one-to-one, but U is
not one-to-one.