EXAM
Exam 3
Math 3351, Spring 2010
April 22, 2011
• This is a Take-home exam.
• Write all of your answers on separate sheets of paper.
Do not write on the Exam questions sheets. You can
keep the exam questions.
• The use of a TI-89 (or similiar) calculator is
expected. State clearly which calculations you are
doing on the calculator.
• You must show enough work to justify your answers.
Unless otherwise instructed,
give exact answers, not
√
approximations (e.g., 2, not 1.414). No Decimals!
• You can use the textbook and your notes. You can
discuss the problems with other people, but write up
your own answers, don’t just copy from someone else.
• This exam has 7 problems. There are 470 points
total.
Good luck!
70 pts.
Problem 1. Consider the matrix


1
1
1
3 −10


 −2 −1
1
0
3 




.
4
2
−2
3
−12
A=




 −4 −2
2
0
6 


−1 −1 −1 −4
12
The RREF of A is the matrix

0
−2

 0 1


R=
 0 0

 0 0

0 0
3
1
0
0
0
0
1


0 −5 


1 −2 
.

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?
70 pts.
Problem 2. You’ll want a calculator for this problem. Consider the vectors








−1
18
−3
4








 0 
 10 
 1 
 4 








v1 = 
.
 , v4 = 
 , v3 = 
 , v2 = 
 1 
 1 
 1 
 1 








1
4
2
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

8




 7 


w1 = 
,
 4 


6
1
−5



 7 


w2 = 
.
 2 


3
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.
80 pts.
Problem 3. Consider the matrix

−9
13
12


10 −12 

15 −19

A=
 −12
−20
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),


−2



x(0) = 
 3 .
−2
40 pts.
90 pts.
Problem 4. Show that the following matrix N is nilpotent. Use this fact to
calculate eN t .


111 −46 −106 11


 −43
21
52 11 


N =

 127 −54 −126
6 


−88
37
86 −6
Problem 5. Consider the matrix

−15

9
A=

−40
−19
2


12 −1 
.
−45
7
This matrix is definitely not diagonalizable! The eigenvalues of A are 1 and 2.
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 .
60 pts.
Problem 6. Let f (x) be the function with period p = 2L = π that is given
on the interval (−π/2, π/2) by
−π/2 < x < π/2.
f (x) = cos(x),
Note that cos(x) has period 2π, not period π, so the periodic function f (x) is
not equal to cos(x) for all x.
Find the Fourier Coefficents of f (x) and the Fourier series.
60 pts.
Problem 7. Consider the function
f (x) = 1 − x,
0 ≤ x ≤ 2.
A. Find the Fourier cosine series of f (x) on the interval [0, 2].
B. What does the cosine series converge to at x = 0? At x = 1? At x = 2?
C. Find the Fourier sine series of f (x) on the interval [0, 2].
D. What does the sine series converge to at x = 0? At x = 1? At x = 2?
Use a calculator to do the integrals in this problem. If all of the even or odd
terms of a series are zero, write the series without these terms, using n = 2k or
n = 2k − 1, k = 1, 2, 3, . . . .
3