Mathematics 244, section 1 – Linear Algebra Exam 3 Sample Questions – April 20, 2007

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

[T]_{std←std} = ( \begin{matrix}1 & 3 & −1 \\ 2 & 0 & 4 \\ 1 & −2 & 1 \end{matrix} )

Let

B = \{( \begin{matrix}1 \\ 1 \\ 1 \end{matrix} ) , ( \begin{matrix}1 \\ 1 \\ 0 \end{matrix} ) , ( \begin{matrix}1 \\ 0 \\ 0 \end{matrix} )\} and C = \{( \begin{matrix}2 \\ −1 \\ 3 \end{matrix} ) , ( \begin{matrix}0 \\ −1 \\ 3 \end{matrix} ) , ( \begin{matrix}2 \\ 1 \\ 0 \end{matrix} )\}

be bases for R^{3}. A) What is the change of coordinates matrix P_{C←std}? B) What is the matrix [T]_{std←B}? C) What is the matrix [T]_{C←std}? D) What is the matrix [T]_{B←C}?

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

III. Let

A = ( \begin{matrix}a & −1 & 0 \\ 0 & 3 & 0 \\ 2 & 2 & −4 \end{matrix} )

A) Find the characteristic polynomial of A. B) Does dimNul(A − (−4)I) depend on the value of a ∈ R? Explain. C) For which values of a ∈ R is the matrix A diagonalizable?

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

T (p(x)) = 2xp′(x) + (x + 1)p′′(x).

A) Show that T is linear. B) Find the matrix [T]_{B←B} with respect to the basis B = {1, x, x^{2}} of P_{2}(R) C) Determine the the eigenvalues of T. D) For each eigenvalue, find a basis for the corresponding eigenspace (express the basis

elements as polynomials). Is T diagonalizable? Why or why not?

V. (True-False) For each true statement, give a short proof. For each false statement, give a reason or a counterexample. A) If A is an n × n matrix whose rows are linearly dependent, then λ = 0 is an eigenvalue

of A. B) If A is an 8 × 8 matrix with det(A − λI) = (2 − λ)^{3}(3 − λ)^{5}, rank(A − 2I) = 6, and

rank(A − 3I) = 4, then it is possible for A to be diagonalizable. C) If the columns of an n × n matrix Q form an orthogonal set in R^{n}, then Q^{t}Q = I,

then n × n identity matrix. D) If S is an orthogonal set of nonzero vectors in R^{n} then S is linearly independent. E) If B is a linearly independent set in the vector space M_{3×4}(R), then B contains at

most 3 + 4 = 7 matrices.