EXAM
Exam 1
Math 5316, Fall 2012
October 8, 2012
• Write all of your answers on separate sheets of paper.
You can keep the exam questions. This is a takehome
exam, to be worked individually. You can use your
notes. You may use Maple as indicated and turn in
hardcopy Maple output with your written answers.
• 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 9 problems. There are 510 points
total.
Good luck!
50 pts.
Problem 1. In each part, you are given a system of linear equations. Form the
augmented matrix of the system and perform row operations to
put the coefficient matrix in RREF. You can use Maple, but show the individual
row operations (check yourself with Maple). Hint: When the left hand side is
the same, you can do several systems at once by forming an augmented matrix
with all of the right-hand sides to the right of the bar.
A.
x1 + 4 x2 + 12 x3 = 2
x1 + 3 x2 + 9 x3 = 2
2 x2 + 7 x3 = −1
B.
3 x2 + 9 x3 − 3 x4 = −15
2 x1 − 4 x3 + 4 x4 = 2
4 x1 − 8 x3 + 8 x4 = 4
−8 x1 + x2 + 19 x3 − 17 x4 = −13
C.
3 x2 + 9 x3 − 3 x4 = 3
2 x1 − 4 x3 + 4 x4 = 0
4 x1 − 8 x3 + 8 x4 = −1
−8 x1 + x2 + 19 x3 − 17 x4 = 4
50 pts.
Problem 2. Consider the linear system whose augmented matrix is


1 2
2 1
 −1 b −3 x 
4 1
1 1
Determine the values of b for which the system has a unique solution. Find
this solution (in terms of x).
For the values of b such that the system does not have an unique solution,
what values of x make the system consistent? Find all solutions of the system
in this case.
1
70 pts.
Problem 3. Consider the matrix


1 −19 −39
55
8 −17
 0
1
2 −3
0
1 



0
5
10
−14
−2
5
A=


 1
0
−1
2
0
2 
5 −98 −201 284 40 −88
The Reduced Row Echelon Form

1
 0

R=
 0
 0
0
on A is
0
1
0
0
0

−1 0
4 2
2 0 −6 1 

0 1 −2 0 

0 0
0 0 
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 column space of A.
D. What is the rank of A?
60 pts.
Problem 4. In each part, use row operations to determine if the given matrix
A is invertible and, if so, find the inverse. Be sure to explicitly
display the inverse matrix.
A. In this part, you can use Maple, but do the row operations one by one.
2 −2
A=
3
2
B. In this part, you can use a Maple command to do the row reduction.


1 2 2
A= 1 1 2 
3 1 4
C. In this part, you can use a Maple command to do the row reduction.


1
0 −3
1 −8 
A= 2
−4 −2 16
2
50 pts.
Problem 5. Consider the vectors





128
439
717

 15 
 23 
4





 , b2 =  60  , b3 =  18
100
b1 = 





 62
 212 
 347 
1
0
4




b4 = 




,


Verify that b1 , b2 , . . . , b5 form a basis of R5 .
Show that the list of vectors




964
1942
 33 
 65 







v1 =  268  , v2 = 
 132  ,
 467 
 942 
0
4

1798
60
247
869
3
1767
59
244
857
4


v3 = 






,




b5 = 


175
6
24
85
0



.





,


is linearly independent and complete it to a basis of R5 by adding on some
vectors from the list b1 , b2 , . . . , b5 .
50 pts.
Problem

−1
 0

v1 = 
 0
 −9
−4
6. Consider the


60
 1



 , v2 =  30


 7

3
vectors



,





v3 = 


62
1
30
25
11




,




v4 = 


−74
−88
−37
−91
0



,





v[5] = 


−326
−179
−164
−185
−1
A. Let S be the the subspace S = span(v1 , v2 , v3 , v4 , v5 ) of R5 . Find a basis
for S. What is the dimension of S?
B. Consider the vectors



w1 = 


−213
−177
−104
−234
−23



,





w2 = 


9
−55
6
−78
−9



.


For each of the vectors w1 and w2 , determine if the vector is in S and, if so,
write it as a linear combination of the basis vectors for S that you found in
the first part.
3






50 pts.
Problem 7. Consider the space
P2 = {ax2 + bx + c | a, b, c ∈ R}
of polynomial of degree less than or equal to 2.
Let
P = 1 x x2
be the usual basis of P2 and let
Q = 41 + 213x + 8x2
10 + 55x + 2x2
5 + 27x + x2
A Verify that Q is a basis of P2 and find the change of basis matrices SPQ and
SQP .
B Let p(x) = 3x2 + x + 2. Find [p(x)]P , the coordinate vector of p(x) with
respect to P. Find [p(x)]Q , the coordinate vector of p(x) with respect to Q.
Write p(x) as a linear combination of the polynomials in Q. Check this result
by high school algebra.
80 pts.
Problem 8. Let
U = u1
u2
be a basis of R2 . Suppose that L : R2 → R2 is a linear transformation such that
L(u1 ) = 2u1 − 3u2
L(u2 ) = −u1 + 2u2
A. Find [L]UU , the matrix of L with respect to the basis U.
B. If v = 2u1 +3u2 , find [v]U and [L(v)]U . Express L(v) are a linear combination
of u1 and u2 .
C. Suppose that
1
u1 =
,
1
2
u2 =
.
1
Find the change of basis matrices SEU and SUE , where
E = e1 e2
is the standard basis of R2 .
D. Find [L]EE , the matrix of L with respect to the standard basis.
4
50 pts.
Problem 9. Let V and W be vector spaces over a field F. Suppose that
L : V → W is a linear map and v1 , v2 , . . . , vn is a basis of V .
Show that L is an isomophism (i.e., L is one-to-one and onto) if and only if
L(v1 ), L(v2 ), . . . , L(vn ) is a basis of W .
5