EXAM
Exam 2
Math 3351, Spring 2010
April 11, 2011
• Write all of your answers on separate sheets of paper,
do not write on the exam questions handout. You
can keep the exam questions when you leave. You
may leave when finished.
• You must show enough work to justify your answers.
• The use of a TI-89 Calculator is expected. State
clearly which computations you are doing on the
calculator. In all cases, give exact answers and
express your numbers as integers or fractions, not in
decimal form (1/3 good, 0.3333 bad).
• This exam has 5 problems. There are 350 points
total.
Good luck!
70 pts.
Problem 1. Consider the matrix


−1
0
2 −1
4
1


 −12 −3
15 −9
2
52 


A=
.

 20
5
−25
15
−1
−89


−4 −1
5 −3
0
18
The RREF of A is the matrix

1 0

 0 1


 0 0

0
0
−2
1
3 −1
0
0
0
0
A. Find a basis for the nullspace of A.
B. Find a basis for the rowspace of A.
C. Find a basis for the columnspace of A.
D. What is the rank of A?
1
0 −5


2 

.
1 −1 

0
0
0
70 pts.
Problem 2. You’ll want a calculator for this problem. Consider the vectors








−1
1
−1
1








 4 
 −3 
 1 
 0 








v1 = 
.
 , v4 = 
 , v3 = 
 , v2 = 
 4 
 −14 
 4 
 1 








1
1
−1
1
Let S = span(v1 , v2 , v3 , v4 ), which is a subspace of R4 .
A. Find a basis of S. What is the dimension of S?
B. Express the vectors in the list v1 , . . . , v4 that are not part of the basis you
found as linear combinations of the basis vectors.
C. Consider the vectors

1




 0 


w1 = 
,
 9 


0
4



 1 


w2 = 
.
 −6 


6
Determine if these vectors are in S. If the vector is in S, express it as a
linear combination of the basis vectors found above.
40 pts.
Problem 3. Let
"
5
−3
6
−4
A=
#
.
Find the characteristic polynomial and the eigenvalues of A, by hand computation. (Do not find any eigenvectors.)
2
80 pts.
Problem 4. Consider the matrix

−10 24

A=
 −9 20
−9 18
−12


−9 
.
−7
This matrix is definitely diagonalizable. The eigenvalues of A are −1 and 2.
A. Find a basis for each of the eigenspaces of A.
B. Find a diagonal matrix D and an invertible matrix P so that P −1 AP = D.
C. Find the matrix exponential etA .
D. Solve the initial value problem
x0 (t) = Ax(t),


1



x(0) = 
 2 .
−2
90 pts.
Problem 5. Consider the matrix


−2
3
1

A=
 −1
2
−4
3

1 
.
3
This matrix is definitely not diagonalizable! The eigenvalues of A are −1 and
2.
A. Find a basis for each of the generalized eigenspaces of A.
B. Find the Jordan decomposition A = S + N of A.
C. Find the matrix exponentials etS , etN and etA .
3