Math 317: Linear Algebra
Practice Exam 2
Spring 2016
Name:
*Please be aware that this practice test is longer than the test you will see on March 11,
2016. Also, this test does not cover every possible topic that you are responsible for on
the exam. For a comprehensive list of all topics covered on the exam, please see the exam
topics document on the website.
I have purposefully strayed away from computation type problems (except for one or two)
to give you some additional insight and practice on the types of proofs you could see on
the exam. The exam is roughly 40% theory and 60% computational, so please be sure
that you know how to compute answers as well! (See Exam Guide topics).
1. Suppose that A is an n×n skew-symmetric matrix and x ∈ Rn satisfies the equation
(A + I)x = 0, where I denotes the identity matrix.
(a) Show that Ax = −x.
(b) Show that xT A = xT .
(c) Show that xT x = −xT x.
(d) Use (a)-(c) to prove that I + A is invertible.
2. Suppose that A is a nonsingular n × n matrix.
(a) Prove that (A−1 )T = (AT )−1 .
(b) Suppose that A is a nonsingular symmetric n × n matrix. Prove that A−1 is
symmetric.
3. Suppose that A and B are n × n symmetric matrices such that AB = BA. Prove
that AB is symmetric. Give an example to show that AB need not be symmetric
if AB 6= BA.
4. Consider the following matrix:


2 2 2
4 7 7  .
6 18 22
Compute the LU factorization of A.
5. Prove that the LU factorization of a matrix A (assuming that it exist) is unique.
Hint: Suppose that A has two LU factorizations, i.e. A = L1 U1 and A = L2 U2 .
Prove that L1 = L2 and U1 = U2 . You may use the fact that if A = LU , then L is
invertible and U is invertible. Furthermore, L−1 is a lower triangular matrix and
U −1 is an upper triangular matrix.
6. Prove or disprove: V = {x ∈ R2 | x1 x2 = 0} is a subspace of R2 .
1
Math 317: Linear Algebra
Practice Exam 2
Spring 2016
7. Suppose that A is a 3 × 3 matrix such that

   
 
1 
 1
 −2 




2 , −1
C(A) = span
, N (A) = span  1  ,




3
2
0

1

and b = −7.
0
(a) Prove that Ax = b is consistent.
(b) Prove that Ax = b does not have a unique solution.
8. Suppose that A is an n × n matrix.
(a) Suppose that A is nonsingular. Prove that C(A) = Rn .
(b) If A is nonsingular, describe its four fundamental subspaces.
9. Let S = {0}.
(a) Prove that S is a linearly dependent set.
(b) Prove that any set containing the zero vector is linearly dependent.
10. Suppose that S = {u1 , u2 , . . . , un } is a linearly independent subset of Rm and that
P is an m × m nonsingular matrix. Prove that P (S) = {P u1 , P u2 , . . . , P un } is a
linearly independent set. Give an example to show why the nonsingularity of P is
necessary.
11. Find a basis and the dimensions of the four fundamental subspaces associated with


1 2 2 3
A = 2 4 1 3 .
3 6 1 4
2