Linear Algebra II, Midterm, Yung-fu Fang, 2010/11/09 Show All Work
1. (a) State the test for diagonalization. [5%]
(b) State the Cayley-Hamilton Theorem. [5%]
(c) State the Gram-Schmidt Process. [5%]
(d) State the Schur Theorem. [5%]
2. Let T : P2(R)→ P2(R) defined by T (f (x)) = f (x) + (x + 1)f′(x). Show that T is diagonalizable
and find the matrices Q and D such that Q−1AQ = D. [10%]
3. Let T : R2 → R2 be the rotation by θ. Prove that T is a linear operator. Is T diagonalizable?
Explain! [10%]
4. Let B1 ∈ Mk×k(F ), B2 ∈ Mk×(n−k)(F ), and B3 ∈ M(n−k)×(n−k)(F ). Show that [10%]
det
( B1− tIk B2 0 B3− tIn−k
)
= det (B1− tIk) det (B3− tIn−k) .
5. Let T be a linear operator on a finite-dimensional vector space V , and let W be a T -invariant subspace of V . Define T : V /W → V/W by T (v + W ) = T (v) + W for any v + W ∈ V/W . Show that if both TW and T are diagonalizable and have no common eigenvalues, then T is diagonalizable. [10%]
6. Let V be a finite-dimensional inner product space with an orthonormal ordered basis β = {v1,· · ·, vn}, T a linear operator on V , and the matrix A = [T ]β. Prove that, for all i and j,
Aij =< T (vj), vi >. Give a direct proof. [10%]
7. Let ∥ · ∥ be a norm on a real vector space V satisfying the parallelogram law,
∥x + y∥2+∥x − y∥2 = 2∥x∥2+ 2∥y∥2. Define
< x, y >= 1 4
[∥x + y∥2− ∥x − y∥2].
Show that
(a) < x, 2y >= 2 < x, y >, for all x, y ∈ V . [5%]
(b) < x + u, y >=< x, y > + < u, y >, for all x, u, y ∈ V . [5%]
8. Let A∈ Mm×n(F ) and b∈ Fm. Suppose that the system of equations Ax = b is consistent.
(a) Prove that R(LA∗)⊥ = N (LA). [5%]
(b) Prove that the minimal solution s to Ax = b is in R(LA∗). [5%]
(c) Find the minimal solution to
x + 2y− z = 1, 2x + 3y + z = 2, 4x + 7y− z = 4.
[5%]
9. Let T be a normal operator on a finite-dimensional real inner product space V whose charac- teristic polynomial splits. Show that V has an orthonormal basis of eigenvectors of T . Hence that T
is self-adjoint. [10%]