Mathematics 244, section 1 – Linear Algebra Exam 3 – April 30, 2004

Directions

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 & 3 & −1 \\ 2 & 0 & 4 \\ 1 & −2 & 1 \end{matrix} )

Find the matrix [T]_{β}^{β}, where

β = {(1, 1, 1), (1, 1, 0), (1, 0, 0)}

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)) = 2xp′(x) + (x + 1)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 whose rows are linearly dependent, then det(A) = 0. B) (10) The vector space M_{2×3}(R) is isomorphic to P_{6}(R). C) (10) If the columns of an n × n matrix Q form an orthonormal 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.