EXAM
Exam 4
Final Exam
Math 2360–102, Summer I, 2015
July 3, 2015
• Write all of your answers on separate sheets of paper.
Do not write on the exam handout. You can keep
the exam questions when you leave. You may leave
when finished.
• You must show enough work to justify your answers.
Unless otherwise instructed,
give exact answers, not
√
approximations (e.g., 2, not 1.414).
• This exam has 8 problems. There are 440 points
total.
Good luck!
40 pts.
Problem 1. In each part, solve the linear system. Use your calculator to find
the RREF, but write down the augmented matrix and the matrix
you wind up with, and then find all solutions.
A.
3x1 + 2x2 + 2x3 = 3
61x1 + 60x2 + 42x3 = 56
x1 + x2 + x3 = 0
7x1 + 7x2 + 6x3 = 3
B.
6x1 + x2 − 9x3 + 13x4 = −3
−4x1 − x2 + 7x3 − 9x4 = 1
19x1 + 5x2 − 34x3 + 43x4 = −4
−13x1 − 4x2 + 25x3 − 30x4 = −1
40 pts.
Problem 2. Consider the matrix


2 1
1 2 
−1 3
3
A= 0
1
A. Find the cofactors C12 , C22 and C32 .
B. Show how to compute det(A), using the cofactor expansion along a selected
row or column.
70 pts.
Problem 3. Consider the matrix

1 15 29

 9 7 5

A=
 3 2 1

0
The RREF of A is the matrix

1

 0

R=
 0

0
43
−14

39 

.
0 13 

15 0
−4
−10
5
10
0
−1
4
3
0 −2
1
2 −2
3
0
0
0
0
0
0
0
0
1

−30
0


0 

.
1 

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?
60 pts.
Problem 4. 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.
Problem 5. Let U = u1
100 pts.
u2 be the basis of R3 given by
"
#
"
#
1
1
u1 =
, u2 =
3
2
Let V be the basis of R3 given by
"
#
1
v1 =
,
−2
"
v2 =
0
#
−1
Let T : R2 → R2 be the linear transformation defined by
T (v1 ) = v1 − 2v2
T (v2 ) = 3v1 + v2
2
A. Find the transition matrices SEU , SEV , SU V and SVU . Recall that E is the
standard basis of R3 .
B. Find the matrix [T ]VV of T with respect to V.
C. Find the matrix [T ]U U of T with respect to U.
D. Let w ∈ R2 be the vector with
"
#
−3
[w]U =
4
Find [T (w)]U .
E. Find [T (w)]V .
50 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


−3
4
4

 −3

−4
4

3 

4
4
and the eigenvalues are 1 and 2.
B. The matrix is
"
A=
−3
20
−1
6
#
and the eigenvalues are 1 and 2.
40 pts.
Problem 7. Consider the three vectors
 
 
1
0
v1 = 0 , v2 = 1 ,
1
1
 
1
v3 = 1
0
Apply the Gram-Schmidt Process to these vectors, in the given order, to produce
an orthonormal basis of R3 .
3
40 pts.
Problem 8. Let S be the subspace of R3 spanned by the orthonormal vectors
 
 
1
1
1  
1  
1 .
u1 = √ 1 , u2 = √
3 1
6 −2
Let w be the vector
 
2
w = 1 .
3
Find the projection of w onto the subspace S.
4