APPM 3310: Matrix Methods — Final — May 2, 2009
On the front of your bluebook print (1) your name, (2) your student ID number, and (3) a grading
table. Explain all of your answers. This test is worth 150 points. There are 20 additional points
available if you choose to answer all the questions. A correct answer with no supporting work may
receive no credit while an incorrect answer with some correct work may receive partial credit. No
books or notes. No electronic devices of any kind (e.g. cell phones, calculators, etc.) are permitted.
Begin each problem on a new page.
1. (30 points) Let P(4) denote the vector space of all polynomials of degree less than or equal to 4.
(a) Are p1(x) = x−2, p2(x) = x2−5x+4, p3(x) = 3x2−4x,p4(x) = x2−1 linearly independent
elements of P(4)?
(b) What is the dimension of V= span{p1, p2, p3, p4}?
(c) Verify the Cauchy-Schwartz inequality for the functions p1=x−2 and p4=x2−1 with
respect to the L2-norm on [0, 1].
2. (40 points) Let K=KTbe a symmetric n×nmatrix.
(a) We learned that Kis positive definite if and only if all of its eigenvalues are strictly positive.
Prove the direction: K > 0 implies all of its eigenvalues are strictly positive.
(b) Prove that K2is positive definite.
(c) Prove that if Kis positive definite then Kcan be written as a Gram matrix.
(d) Find the Gram matrix Kfor the monomials 1, x,x2under the L2inner product on [0,1].
3. (50 points) For this problem let
A=
1−1 0
−1 2 −1
0−1 1
(a) Find the eigenvalues and eigenvectors of A.
(b) Is Apositive definite, positive semi-definite or neither? Explain.
(c) Find orthonormal eigenvector bases for each of the four fundamental subspaces of A.
(d) Find and orthonormal basis for R3consisting of eigenvectors of A. Verify orthonormality
for full credit.
(e) Write out the spectral factorization of A.
4. (40 points) A few unrelated short answer questions.
(a) State the Fundamental Theorem of Linear Algebra parts 1 and 2. (Hint: part 1 characterizes
the dimensions of the four fundamental subspaces of a matris; part 2 gives orthogonality
relations between them.)
(b) Give the definition for Wto be a subspace of a vector space V.
(c) State how positive definite matrices are related to inner products on Rn.
(d) Suppose the vector space V=span{v1,v2, ...vn}. What do you know about dim(V)?
5. (10 points) For the quadratic form q(x) = 2x2
1+x1x2−2x1x3+ 2x2
2−2x2x3+ 2x2
3, find the
vector v∈R3that minimizes q(x). (Hint: use a matrix to represent q(x).)