Mathematics 244, section 1 – Linear Algebra Exam 3 – April 27, 2007

Directions

Do all work in the blue exam booklet – there are 100 possible points, and 10 possible Extra Credit points (see question III).

I. Let V = Span{e^{x}, cos(x), sin(x)}, let W = P_{2}(R) and let T : V → W be the mapping defined by

T (f) = f(0) + f′(0)x + \frac{f′′(0)}{2}x^{2}.

A) (10) Show that T is a linear mapping. B) (15) Let B = {e^{x}, cos(x), sin(x)} (a basis for V) and C = {1, x, x^{2}} (basis for W).

Find the matrix [T]_{C←B}. C) (15) Now, let C′ = {x^{2} + x, x − 2, x + 1}. Find the matrix [T]_{C′←B}.

II. Let

A = ( \begin{matrix}1 & s & s \\ −1 & s & s \\ −1 & −1 & 1 \end{matrix} )

The characteristic polynomial of A is

p(λ) = −λ^{3} + (s + 2)λ^{2} − (5s + 1)λ + 4s.

A) (10) Determine a value of s such that λ = 3 will be an eigenvalue of A. B) (20) Let s = −1. Is A diagonalizable in this case? If so, find an invertible matrix Q

and a diagonal matrix D such that Q^{−1}AQ = D. If not, say why not. Note: You may “purchase” a factored form of the characteristic polynomial with s = −1 from me during the exam for a “price” of 5 points on this part.

III. (True/False) For each true statement, give a short proof. For each false statement, give a counterexample. Do any three parts. If you submit solutions for all four, I will count any additional points you earn as extra credit.

A) (10) Let A be an n × n matrix. If {v_{1}, …, v_{k}} are eigenvectors of A with distinct

eigenvalues λ_{i}, i = 1, …, k, then {v_{1}, …, v_{k}} is an orthogonal set. B) (10) If A and B are n × n matrices with the same characteristic polynomial, then A

and B are similar. C) (10) If v is a nonzero vector in R^{n}, and P_{v} is the orthogonal projection mapping to

the line spanned by v, then the matrix [P_{v}]_{B←B} has rank 1 no matter which basis B is used in R^{n}.

D) (10) If a 6×6 matrix A has characteristic polynomial (1−λ)^{3}(6−λ)^{3}, and rank(A−I) = 4 and rank(A − 6I) = 3, then A is diagonalizable.