PROBLEM SET
Problems on Span, Independence, and Matrix Spaces
Math 3351, Fall 2010
Sept. 27, 2010
• Write all of your answers on separate sheets of paper.
You can keep the question sheet.
• You must show enough work to justify your answers.
Unless otherwise instructed,
give exact answers, not
√
approximations (e.g., 2, not 1.414).
• This problem set has 8 problems.
Good luck!
Problem 1. Consider the matrix

−15 −80 −145

 −89 −19
51

A=
 66 −62 −190

77
81
85

22
−181
50

132 

.
−96 

150
78
−8
A Find a basis of the nullspace of A.
B Find a basis of the rowspace of A.
C Find a basis of the column space of A. Write the other columns of A as linear
combinations of these basis vectors.
D What is the rank of A?
Problem 2. Consider the matrix

8 9

 7 1


 5 5

A=
 2 7


 4 4

4 1

19
4
22 −25
−4
6
10
2
19
4
8
6
−1
6

10 


13 −12 


12 −33 


6 −14 

0
1
3
A Find a basis of the nullspace of A.
B Find a basis of the rowspace of A.
C Find a basis of the columnspace of A. Write the other columns of A as linear
combinations of these basis vectors.
D What is the rank of A?
1
r
Problem

5

 6

v1 = 
 2

3. Consider the vectors





11
7





 12 
 8 






,
 , v3 = 
 , v2 = 
 −1 
 1 






12
4
0

12



 14 


v4 = 
,
 3 


4

3



 3 


v5 = 
.
 8 


1
Let S = span(v1 , v2 , v3 , v4 , v5 ), 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 , . . . , v5 that are not part of the basis as
linear combinations of the basis vectors.
C. Consider the vectors

7




 9 


w1 = 
,
 0 


3
12



 13 


w2 = 
.
 27 


−1
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

 6

v1 = 
 2

4. Consider the vectors





7
11






 8 
 12 





 , v2 = 
 , v3 = 
,

 1 
 −1 





0
4
12

12



 14 


v4 = 
,
 3 


4

3



 3 


v5 = 
.
 8 


1
Let S = span(v1 , v2 , v3 , v4 , v5 ), which is a subspace of R5 .
A. Find a basis of S. What is the dimension of S?
B. Express the vectors in the list v1 , . . . , v5 that are not part of the basis as
linear combinations of the basis vectors.
2
C. Consider the vectors

7




 9 


w1 = 
,
 0 


3
12



 13 


w2 = 

 27 


−1
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. Determine if the following vectors are independent or dependent.
If they are independent, complete the list to a basis of R5 by
adding on a standard basis vector. If the vectors are dependent, find a linear
relation between them.








1
−1
4
13








 −2 
 1 
 −1 
 −10 
















 , v2 =  1  , v3 =  0  , v4 =  8 
24
v1 = 
















 0 
 2 
 −2 
 −9 








−4
0
0
−11
Problem 6. Determine if the following vectors are independent or dependent.
If they are independent, complete the list to a basis of R5 by
adding on a standard basis vector. If the vectors are dependent, find a linear
3
relation between

−38

 91


v1 = 
 −1

 63

−23
them.





,





−63
1

4

 5

v5 = 
 1

3
12




 45 





v3 =  −14 
,


 60 


−35


 −26 





v2 =  30 
,


 10 


22
Problem 7. Consider the vectors


 
4
4


 
 2 
 5 


 
v1 = 
 , v2 =   ,
 0 
 3 


 
1



4



 −1 


v3 = 
,
 −3 


1



0




 6 



 , v6 = 


 0 



6
−89



 −170 





v4 =  64 



 24 


42

28



 26 


v4 = 
,
 12 


7
Do these vectors span R4 ? If so, select a basis of R4 out of this list of vectors.
4
Problem 8. Consider the vectors
 


9
3
 


 4 
 6 
 


v1 = 
 , v2 =   ,
 2 
 5 
 



3



 −8 


v3 = 
,
 −8 


−12
4
8





3
9






 4 
 10 





v5 = 
 , v7 = 
 , v6 = 

 8 
 0 





0
10

3



 0 


v4 = 
,
 7 


2

42

22 


47 

8
Do these vectors span R4 ? If so, select a basis of R4 out of this list of vectors.
5