EXAM
Exam 3
Math 2360, 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 8 problems. There are 500 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
−2
1
3 −1
0
0
0
0
0
0 −5


2 

.
1 −1 

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
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.
1
80 pts.
Problem 3. Recall that the standard basis of R2 is E = e1
1
0
e1 =
, e2 =
.
0
1
e2 where
Let
U = u1
where
u1 =
2
,
2
u2 ,
u2 =
3
,
2
and let
V = v1
where
4
v1 =
,
3
v2 ,
5
v2 =
.
4
Then U and V are ordered bases of R2 (you don’t need to check that).
A. Find the change of basis matrices SEU and SEV .
B. Find the change of basis matrices SU V and SVU .
C. Let w ∈ R2 be the vector such that
[w]U =
3
.
−2
Find [w]E and express w as an element of R2 .
D. Find [w]V , the coordinate vector of w with respect to V.
60 pts.
Problem 4. Recall that the standard basis of R2 is E = e1
1
0
e1 =
, e2 =
.
0
1
Let
U = u1
where
2
u1 =
,
2
u2 ,
3
u2 =
,
2
and let
V = v1
2
v2 ,
e2 where
where
v1 =
4
,
3
v2 =
5
.
4
Then U and V are ordered bases of R2 (you don’t need to check that). You can
use the results of the previous problem without repeating the calculations here.
Let L : R2 → R2 be the linear transformation such that
L(u1 ) = 2u1 − 3u2
L(u2 ) = −3u1 + 2u2 .
A. Find [L]U U , the matrix of L with respect to the basis U.
B. Find [L]VV , the matrix of L with respect to the basis V.
C. Let w ∈ R2 be the vector whose coordinate vector with respect to V is
−2
[w]V =
.
4
Find [L(w)]V , the coordinate vector of L(v) with respect to the basis V.
40 pts.
Problem 5. Let
"
4
−1
6
−1
A=
#
.
Find the characteristic polynomial and the eigenvalues of A by hand computation. (Do not find any eigenvectors.)
60 pts.
Problem 6. In each part, you are given a matrix A and its eigenvalues. Find
a basis for each of the eigenspaces of A and determine if A is
diagonalizable. If so, find a diagonal matrix D and an invertible matrix P so
that P −1 AP = D.
A. The matrix is

13 −9

A=
 −12
−20
12


10 −12 
.
15 −19
and the eigenvalues are 1 and 2.
B. The matrix is

−13
47
−8

A=

−4
15
−2
6

−4 

2
and the eigenvalues are 1 and 2.
3

C. The matrix is

−592
1862

A=
 −186
585
−57
178
89


28 

12
and the eigenvalues are 1, 2 + 3i and 2 − 3i.
Instructions: Use a calculator the find RREFs and to do matrix algebra.
Otherwise, do the work by hand. (You can’t use the eigenvalue and eigenvector
functions on the calculator, which only give approximate answers in any case.)
60 pts.
Problem 7.
A. Consider the three vectors
 
1
 

v1 = 
 2 ,
0

1




v2 = 
 0 ,
1

1




v3 = 
 1 ,
1
which are a basis of R3 . Apply the Gram-Schmidt Process to these vectors
to produce an orthonormal basis of R3 .
B. Let w be the vector
 
1
w = 0 .
0
Find the projection of w onto the subspace of R3 spanned by v1 and v2 .
60 pts.
Problem 8.
A. Let S1 be the subspace of R5 defined by the equations
x1 − x2 − 3x3 + 2x4 + x5 = 0
33x1 + 16x2 − x3 + 74x4 + 188x5 = 0
4x1 + 2x2 + 9x4 + 23x5 = 0
69x1 + 32x2 − 5x3 + 154x4 + 388x5 = 0.
Express S1 as the nullspace of some matrix A. Find a basis for S1 .
B. Let S2 be the subspace of R4 spanned by the vectors
 
 
 
1
0
2
12
9
8

 
 
v1 = 
10 , v2 =  9  , v3 =  4  .
27
18
22
4
Express S2 as Range(B) for some matrix B. Find a set of equations that
defines S2 . [Hint: Find a basis for the nullspace of B T .]
5