Math 215 Section 2 Third Hour Exam December 6, 1995

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The number in parentheses before each question gives its point value. The total is 100 points. If you want any partial credit, you must show your work.

1. (15) Let A be the matrix ( \begin{matrix}2 & −4 & 2 \\ 0 & −3 & 0 \\ 0 & −10 & 2 \end{matrix} ).

a) Calculate the characteristic polynomial of A.

b) Find the eigenvalues of A, with their multiplicities

c) For each eigenvalue, find a basis of the corresponding space of eigenvectors.

2. (15) True of False?

a) Let A = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ). There exists a 2 × 2 real matrix P such that P AP^{−1} is diagonal.

b) A square matrix A is invertible if and only if 0 is not an eigenvalue of A.

c) If A is a square matrix which has 3 as an eigenvalue with multiplicity 2, then there exist two linearly independent eigenvectors for A.

d) If v and w are non-zero vectors in R^{n}, A is an n × n matrix, Av = 2v, and Aw = 3w, then v and w are linearly independent.

3. (15) Let v = ( \begin{matrix}2 \\ 3 \end{matrix} ) and w = ( \begin{matrix}1 \\ 1 \end{matrix} ).

a) What is the angle between v and w?

b) Write v as the sum of a vector in Span(w) and a vector in w^{⊥}.

c) What is the distance between v and the line through w and the origin?

4. (20) Let V ⊆ R^{3} be the space of solutions of the equation 2x_{1} − 3x_{2} + x_{3} = 0. a) Find a basis of V.

b) Apply the Gram-Schmidt process to your basis in part a to obtain an orthonormal basis of V.

5. (20) Let P_{3} be the vector space of polynomials of degree at most 3 in the variable x. Define a linear transformation T : P_{3} → R^{3} by the formula

T (f(x)) = ( \begin{matrix}f(−1) \\ f(0) \\ f(1) \end{matrix} ) .

a) Compute the matrix of T with respect to the basis B = {1, x, x^{2}, x^{3}} of P_{3} and the standard basis of R^{3}.

b) Is T onto?

c) Is T one to one?

6. (15) Find an orthonormal basis of R^{2} consisting of eigenvectors of the matrix ( \begin{matrix}1 & 2 \\ 2 & 1 \end{matrix} ).