Math 237-101 (CRN 30074)
Carter
Final
Summer 2015
General Instructions. Do not write on this test sheet. Do all your work inside your
blue book. Write neat, complete solutions to each of the problems below. When asked for a
specific calculation, check your results! If your result seems incorrect and you do not have
time to find an arithmetic mistake, then indicate that you know there is a mistake.
According to folk-lore, watermelon season coincides with when crepe-myrtles are in
bloom.
1. (5 points) State the Rank-Nullity Theorem.
2. (5 points) Consider a matrix A ∈ M (7, 9) (seven rows and nine columns). Suppose
that the rank of A is 3. What is the dimension of the null space? Is the linear map
that is represented by A with respect to the standard basis surjective (onto)?
3. (10 points) Write the general solution set to the equation
x + 2y + 3z + 5w = 30
in vector form.
4. (10 points) Solve the system of equations: Consider the system:
x
+ y + z = 1
2x
+ z = 4
y + z = 2
5. The reduced row echelon form of the matrix A is given as indicated.




1
3 4 −2 3
1 0 0 0 − 23
14




 −1 11 6 4 4 
 0 1 0 0 −9 
7 




 rref 
67 
A= 2
−→


1 3 −3 1
0 0 1 0 28 




 1

 0 0 0 1 15 
1
1
1
0



28 
4 −3 2 −2 1
0 0 0 0 0
(a) (5 points) Give a basis for the row space of A.
(b) (5 points) Give a basis for the column space of A.
(c) (5 points) Give a basis for the null space of A. Be aware! This is not an
augmented matrix.
1
6. (15 points) Find a steady state vector (a vector x = [x, y, z]t such that Ax = x with
x + y + z = 1) for the matrix


0.80 0.05 0.05


.
A=
0.10
0.90
0.10


0.10 0.05 0.85
7. (10 points) Given an LU factorization of the matrix


−2 −2
1


.
A=
−6
0
4


2 −2 −1
8. (15 points) Compute the matrix product.

1 0
0
 
1
1 0
 
1 −2
1
1 −2


 
 

 0 1 −1  ·  0 1 0  ·  0 2 −1 1 4 

 
 

2 −3 1 1 −2
−2 0 1
0 1 −2
9. (15 points) Solve the system of equations:
x
− 2y + z + w = −2
2y − z + w =
4
2x − 3y + z + w = −2
10. (15 points) The eigenvalues for the matrix


3 −1 1



A=
−2
4
2


−1 1 5
are λ = 2, 4, 6. Find the corresponding eigenvectors.
A
11. (10 points) Consider the linear mapping R2 ←−R4 denote the linear mapping that is
given by the matrix expression

"
Ax =
1 −2
0
2
x
#  
 y 

.
 z 
−3 2
 
t
0
2
Find a basis for the null space (kernel) of the matrix A.
2

12. (10 points) Determine if the set
     



 1   2   0 







S =  1 , 1 , 1 




 2
3
1 
is linearly independent. Does it span R3 ?
13. (10 points) Compute the determinant of


3 −1 1


.
A=
−2
4
2


−1 1 5
14. Let
"
A=
7
4
4 13
#
.
(a) (5 points) Determine the eigen-values and associated eigen-vectors.
(b) (5 points) Find matrices P and D such that D is diagonal, P is invertible and
P −1 AP = D.
(c) (5 points) Compute A5 .
3