APPM 3310: Matrix Methods
Final Exam — April 30, 2011
On the front of your bluebook print (1) your name, (2) your student ID number, and (3) a grading
table. Show all work in your bluebook. Start each problem on a new page. A correct answer
with no supporting work may receive no credit while an incorrect answer with some correct work
may receive partial credit. One page of notes is permitted, but no other books or electronic devices
are allowed. Sign your bluebook under the Honor Co de to indicate that you have neither
given nor received unauthorized assistance on this exam.
Do problems 1, 2 and 3. Then, choose two of the four problems 4-6 on page 2. Indicate which
problem you are skipping by putting an X through that number on your grading table.
1. (40 Points) For this problem, use the matrix and vector
A=
010
001
1−4 4
,b=
1
1
1
(a) Is Aregular or not? If it is regular, find its LU decomposition. If not, find its permuted
LU decomposition.
(b) Does Ax=bhave a solution for every b? Why or why not? Using Gaussian elimination,
find the solution for the given b.
(c) What is the characteristic polynomial pA(λ) of A?
(d) Show that λ= 1 is an eigenvalue of A. What is the eigenvector for this λ?
(e) Define a complete matrix. Is Acomplete?
2. (40 points) State the fundamental theorem of linear algebra. For each property given
below, give an explicit matrix Awith that property or explain why no such matrix exists.
(a) rng(A) = span2
1, and corng(A) = span
1
2
−1
.
(b) The vector
1
−1
2
is in the kernel of Aand
1
2
1
is in the corange of A.
(c) ker A={0}and coker(A) = span1
2.
(d) Ahas an eigenvalue −1 with multiplicity two, but only one eigenvector.
(e) Ais real, symmetric, and has eigenvalues 2 + iand 2 −i.
3. (30 points) Let S={p(x)∈P(2) |p(1) = 0}, that is the space of quadratic polynomials that
vanish at x= 1.
(a) Define vector subspace.
(b) Show that Sis a vector subspace of P(2).
(c) Define the L2inner product on the interval (0,2).
(d) Find an orthogonal basis for Susing the inner product in (c).
(e) What is dim(S)? What is the dimension of S⊥, the orthogonal complement of Sin P(2) ?
