Mathematics 244, section 1 – Linear Algebra Makeup Exam 3 – May 6, 2004

Directions

Read each question carefully. Do all work in the blue exam booklet. There are 100 regular points and 10 Extra Credit points. Notation:

• P_{n}(R) is the vector space of polynomials of degree ≤ n.

• P (R) is the vector space of all polynomials (no restriction on degree).

• M_{m×n}(R) is the vector space of all m × n matrices.

I. (25) Let T : R^{3} → R^{3} be the linear mapping whose matrix with respect to the standard basis is

A = ( \begin{matrix}1 & 2 & −3 \\ 3 & 2 & 0 \\ 0 & −1 & 12 \end{matrix} )

Find the matrix [T]_{β}^{β}, where

β = {(0, 0, 1), (0, 1, −1), (1, −1, 1)}

II. (20) Let V be a finite-dimenional vector space, let T : V → V be a linear mapping. Show by mathematical induction that if x_{i} ∈ V are vectors satisfying T (x_{i}) = λ_{i}x_{i} for i = 1, …, n, where λ_{i} are distinct real numbers, then {x_{1}, …, x_{n}} is linearly independent.

III. Let T : P_{2}(R) → P_{2}(R) be the linear mapping defined by

T (p(x)) = 2p(x) + (x − 2)p′(x).

A) (20) Find the matrix of T with respect to the basis {1, x, x^{2}} of P_{2}(R) and the eigenvalues of T. For each eigenvalue, find a basis for the corresponding eigenspace (express as polynomials).

B) (5) Is T diagonalizable? Why or why not?

IV. (True-False) For each true statement, give a short proof. For each false statement, give a reason or a counterexample. A) (10) If A is an n×n matrix with det(A) = 0, then the rows of A are linearly dependent. B) (10) The vector space M_{3×2}(R) is isomorphic to P_{5}(R). C) (10) If the columns of an n × n matrix Q form a basis for R^{n}, then Q^{t}Q = I_{n}.

Extra Credit (10) If a, b, c are distinct real numbers, show that the matrix

A = ( \begin{matrix}1 & a & a^{2} \\ 1 & b & b^{2} \\ 1 & c & c^{2} \end{matrix} )

is invertible.