2406
FINAL MATH EXAM
NAME: December 6, 2004
Answer the following questions clearly and completely. PROVIDE PROOF, WORK, OR EXPLANATION FOR EACH PART. There are 55 points total, plus 5 points extra credit. NOTE: Throughout this exam, you can assume that scalars are real. 1. Suppose that V is a 3-dimensional vector space and that T : V V is a linear transformation. Let B = {v1 , v2 , v3 } be a basis for V . Suppose that the matrix for T with respect to the basis B is the matrix 1 2 0 A = [T ]BB = 2 4 0 . 0 0 5 (3 points) a. Prove that x span{v1 , v2 } if and only if T (x) span{v1 , v2 }. (3 points) b. Let x V . Prove that x N (T ) if and only if [x]B N (A). (3 points) c. Find a basis for the nullspace of T . (3 points) d. Prove that v3 is an eigenvector of T . (3 points) e. Let be the eigenvalue for v3 . Find a basis for the -eigenspace. (3 points) f. Determine if T is diagonalizable. If so, give a basis for V consisting of eigenvectors of T .
2. For each of the following statements, either prove is it true or show a counterexample. (3 points) a. If A, B are n n matrices then det(A + B) = det(A) + det(B). (3 points) b. If A, B are n n matrices then det((A + B)2 ) = (det(A + B))2 . (3 points) c. If A is an n n upper triangular matrix then A is invertible. (3 points) d. If A is an n n invertible matrix then det(A1 ) = 1/ det(A). (3 points) e. If A is an n n diagonalizable matrix then it is invertible.
(5 points) 3. Suppose that A and C are m n matrices with m > n. Suppose that C = BA where B is an m m invertible matrix. Prove the that columns of C are independent if and only if the columns of A are independent. Note that A and C are not square matrices.
1
4. Let P3 = {p(x) = ax3 + bx2 + cx + d : a, b, c, d R} be the vector space of all polynomials of degree at most 3. For this problem, the norm of a polynomial p and the inner product of two polynomials p and q will be dened as
1 1/2 1
p
=
0
|p(x)| dx
2
and
p, q =
0
p(x) q(x) dx.
The distance between p and q is p q . Two polynomials p and q are orthogonal if p, q = 0. (3 points) a. Find the orthogonal projection of the polynomial q(x) = 3x2 onto the line spanned by the constant polynomial p(x) = 1. (3 points) b. Let W = span{1, 3x2 }. Find an orthonormal basis for W . (3 points) c. Find the polynomial in W that is closest to the polynomial 4x3 + 1. (3 points) d. Find a polynomial in span{1, 3x2 , 4x3 + 1} that is orthogonal to W .
(5 points) 5. Suppose that A is an n n matrix such th...