Math 323
Exam 2 Sample Problems
October 27, 2013
The exam will cover material from class, the homework, and the reading. Included below are
some sample problems, which should give you a sampling of what types of things to expect
on the exam. One thing though, this is not meant to be an exhaustive list, and there may
be types of problems on the exam that differ from these.
1. Determine if the following sets of vectors are or are not linearly independent. Explain
how you know a set is or is not linearly independent. If a set is linearly dependent,
then find a linear dependency among the vectors.
    
1 
 3



4 , 0
(a)


5
9
        
1
−1
1 
 3
(b) 4 , 0 ,  2  , 1


5
9
0
1
        
1
2
1
−1 


        

3   2  −3  0 
(c) 
,
,
,
−4 −4  2   1 





2
0
−4
0
2. For (a)–(c) in problem 1, find bases for the spans of the given sets of vectors.
3. Given
1
v1 =
,
2
v2 =
−2
,
−3
S=
4 −5
,
−5 −6
find vectors w1 and w2 so that S is the transition matrix from {w1 , w2 } to {v1 , v2 }.
n o
n o
1
1
0
1
0
0
1 , 0 , 1
0 , 1 , 0
be the standard basis of R3 . Let A =
4. Let S =
0
0
1
0
1
1
n 0 o
1
1
1
1 , 0 ,
and B =
.
−1
0
1
(a) Show that A is a basis for R3 . Show that B is not a basis for R3 .
(b) Find a linear equation (or equations) whose set of solutions is Span(B).
3
(c) Find the coordinates of 4 with respect to the basis A. (You may want to do
5
the next part first.)
(d) Find the change of basis matrix which converts coordinates in the S-basis into
coordinates in terms of the A-basis.
5. Let


5 −4 1 8
A = 1 −2 1 2 .
3 0 −1 4
(a) Find bases for the row space and the column space of A.
(b) What is the rank of A?
(c) What is the nullity of A?
c
(d) Let b = 86 . For what value or values of c will Ax = b be consistent?
6. Let A and B be n × n matrices, and let O be the n × n zero matrix.
(a) Prove that AB = O if and only if the column space of B is a subspace of the null
space of A.
(b) Show that if AB = O, then rank(A) + rank(B) ≤ n.
7. Consider the function L : R3 → R2 defined by
 
a
2a
+
6b
L  b  =
.
3c − 9a
c
(a) Is L a linear operator? Explain.
(b) Find a matrix A so that L(x) = Ax for x ∈ R3 .
(c) What is the dimension of the kernel and image of L. Explain.
8. Let T : R2 → R2 be defined by
T (x) =
2 1
x,
1 −3
where x ∈ R2 is written with respect to the standard basis. Let B = {( 11 ) , ( 12 )}. What
is the matrix representing T with respect to the basis B? (I.e., what is [T ]BB ?)
9. Suppose that A and B are similar n × n matrices. Let c ∈ R, and let I denote the
n × n identity matrix. Show that A − cI and B − cI are similar matrices.
10. For each of the following matrices, find the eigenvalues and determine the corresponding
eigenspaces.
1 3
(a)
4 2


1 1 −1
(b) 2 0 1 
1 1 0