1. [9 pts] Prove or disprove the following statements.
(a) Let V = C
4 and let S1 and S2 be two nonempty subsets of V . Then S1 = S2 if and only if S
⊥
1 = S
⊥
2
.
(b) The subset U = {v ∈ R
3
: kvk ≤ 12} is a subspace of R
3
.
(c) Let V = P3(R). Any linear operator on V whose eigenvalues are 1, 2, 3, and 4 is diagonalizable.
2. [8 pts] Let W be an inner product space and suppose T is a linear operator on W with adjoint T
∗
.
(a) Prove that if T
∗T = 144 · IW , then kT(x)k = 12kxk for all x ∈ W.
(b) Suppose W is not the zero vector space. Is it possible for T
∗T = 4i · IW ? Explain your answer.
3. [14 pts] Let β = {1, x, x2} denote the standard ordered basis for P2(R). Suppose T : P2(R) → P2(R) is linear
map with
[T]β =
−2 1 0
0 −2 0
0 0 7
.
(a) Compute T(f(x)) where f(x) = 1 − 3x. Is f(x) an eigenvector for T? Explain your answer.
(b) Is (1, 0, 0) an eigenvector of T? Explain your answer.
(c) Compute the characteristic polynomial of T. Then compute the eigenvalues of T and their algebraic
multiplicities.
(d) Determine if T is diagonalizable.
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