Math 237
Carter
Study Guide for Final
Fall 2010
General Information. This study guide consists of two parts. In the first, I will give
a large number of review problems. These include definitions and theorem statements from
which I will select questions. In addition, I will cull problems from the assigned homework
and other sources so that you may focus your studies. The second section consists of verbatim
tests from this semester, and Spring 2009. My goal is for you to be prepared for the final
exam. According to my reading of the final exam schedule:
http://www.southalabama.edu/registrar/dates.htm
the final will be held from 8AM through 10AM on Monday Dec 13, 2010.
Required Definitions
1. The span of the set {v1 , v2 , . . . , vk } ⊂ V is the set
{v ∈ V : v =
k
X
αi vk for some αi ∈ R}
i=1
2. If the homogeneous equation
x1 v1 + x2 v2 + . . . xk vk = 0
has only the trivial solution x1 = x2 = . . . = xk = 0, then the set {v1 , v2 , . . . , vk } ⊂ V
is said to be linearly independent.
3. 
The row
 space of a matrix A ∈ M (m × n) is the span of the set of rows. Say A =
A1


 A2 
Pm


 .  . Then Row(A) = i=1 yi Ai where yi ∈ R.
 .. 


Am
4. Let A ∈ M (m, n) denote an (m × n)-matrix. Say A = [A1 , A2 , . . . , An ]. The column
space of A is the span of the set {A1 , . . . , An }; i.e.
Col(A) =
n
X
j=1
where xj ∈ R.
1
xj A j
5. Let A ∈ M (m, n) denote an (m × n)-matrix. Say A = [A1 , A2 , . . . , An ]. The null space
of A is the set
Null(A) = {X ∈ M (n, 1) : AX = 0}.
6. A basis for a vector space V is a set {v1 , v2 , . . . , vk } ⊂ V such that
(a) {v1 , v2 , . . . , vk } spans V in the sense that any vector v ∈ V , can be written as
P
v = ki=1 αi vk for some scalars αi ∈ R, and
(b) the set {v1 , v2 , . . . , vk } ⊂ Rn is linearly independent (see above).
7. If the vector space V has a basis {v1 , v2 , . . . , vk }, then the dimensions of V is k, that
is, the number of elements in a linearly independent set that spans V .
8. A vector X ∈ M (n, 1) is said to be an eigenvector for the matrix A if there is a real
number λ 6= 0 such that AX = λX. The number λ is said to be an eigenvalue for the
matrix A.
Basic Theorems
Be able to state these precisely.
Theorem 1. Let A be an (m × n)-matrix. Then
Rank(A) + dim(Null(A)) = n.
Theorem 2. Let A ∈ M (n, n). Then the following are equivalent.
1. A is non-singular;
2. rank(A) = n;
3. dim(Null(A)) = 0;
4. Null(A) = {0};
5. the equation AX = 0 has only the trivial solution;
6. for any B ∈ M (n, 1), the equation AX = B has a solution;
7. for any B ∈ M (n, 1), the equation AX = B has a unique solution;
8. the columns of A are linearly independent;
2
9. the rows of A are linearly independent;
10. the columns of A span M (n, 1);
11. the rows of A span M (1, n);
12. det(A) 6= 0.
Review the homework handouts. Make sure that you can do all of the problems on these.
Page 15-20
# 5, 10, 14, 23
Page 36
# 7, 8
Page 57-63
# 3, 5
Page 80
# 1, 14
Page 86
# 27
Page 106
# 1, 3
Page 120
# 10-13, 19, 20
Page 161
# 6,7, 11, 20
Page 183-187 # 9, 20 (Think about eigenvalues), 37
Page 198
# 3, 10, 11, 20, 26
Page 235
# 4,6
Page 252
#1
Page 282
# 3, 6, 7
1. Determine the equation of the plane whose solution space is the span of the set of
vectors {[1, 1, 2]t , [1, 2, 0]t }.
2. Determine bases for the column space, row space, and null space of the matrix


1 2 −1 3


.
A=
2
2
−4
4


1 3 0 4
3. Determine (draw) the image of the unit circle under the linear transformation
"
# that is
1 1
represented (with respect to the standard basis) by the matrix L =
.
0 1
3
4. For each of the following matrices, find the characteristic equation, the eigenvalues,
and corresponding eigenvectors and eigenspaces, and give a matrix P such that the
matrix P −1 AP is diagonal, or explain why this is not possible.
4 0
(a) A =
5 1


3 3 2


(b) A =  0 −1 2 
0 0 1


0 0 0


(c) A =  2 0 0 
−1 4 0
Old Tests
Math 237
Carter
Test 1 Spring 2009
General Instructions: Do not write on this test. Write your name on only the outside
of your blue books. Do all your work inside your blue books. Write neat complete answers
to the questions below. Show your work.
Remember to use your turn signals when changing lanes. Also observe the person slightly
in front of you with the turn signal blinking probably wants to change lanes; yielding to that
person will not likely slow you down. Good luck.
1. Define the italicized terms:
(a) The set {A1 , A2 , . . . , Ak } is linearly independent
(b) The set {A1 , A2 , . . . , Ak } spans a vector space V
(c) The null space of an (m × n)-matrix
(d) The span of a set of vectors {A1 , A2 , . . . , Ak }
2. Give the matrix A whose (i, j)th entry aij is given by aij = 2j − i for 1 ≤ i ≤ 3 and
1 ≤ j ≤ 3.
3. Give an example of a linearly dependent set of 3 vectors in M (2, 1) such that any two
vectors taken from the set forms a linearly independent set.
("
# "
# "
#)
1 2
0 1
2 2
4. Determine if the set
,
,
is linearly independent or not.
0 1
1 1
1 1
4
5. Give an example of a system of 3 equations in 3 unknowns for which the solution set
is a plane.
6. Solve the system of equations
3x + 7y + 2z = 1
x
−
y
+
z
= 2
5x + 5y + 4z = 5
7. Is the matrix

0 1 2 2 4




A=
0
0
1
2
4


0 0 0 0 1
in reduced row echelon form? If not, then row reduce it. Solve the system of equations
AX = ~0. Give spanning sets for the row space, column space, and null space for A.
8. Assume that the matrix C given below is an augmented matrix: C = [A|B] for some
A and B so

0 1 2 2 4



.
C=
0
0
1
2
4


0 0 0 0 0
Write the general solution to the system of equations AX = B.
Math 237
Carter
Test 2 Spring 2009
General Instructions: Do not write on this test. Write your name on only the outside
of your blue books. Do all your work inside your blue books. Write neat complete answers
to the questions below. Show your work.
While popping pop-corn on the stove, you should heat the oil and throw in two or three
kernel. When the test kernels pop, you can add the rest. Cover the pan, but lift the top
slightly and occasionally to let the steam escape. Be careful not to burn the kernels, and try
not to leave too many uncooked seeds. Remove from heat immediately. Rely on your ears
and nose to tell you when it is done. Good luck.
1. Let A be a given (4×3)-matrix. Explain in complete sentences why there is an equation
AX = B which has no solution.
2. Determine the equation of the plane whose solution space is the span of the set of
vectors {[1, 1, 2]t , [1, 2, 0]t }.
5
3. Show that the set





a
b
c








U =  0 d e  : a, b, c, d, e, f ∈ R





 0 0 f
of upper triangular (3 × 3)-matrices is a subspace of the space M (3, 3). What is the
dimension of U?
4. Determine bases for the column space, row space, and null space of the matrix


1 2 −1 3


.
A=
2
2
−4
4


1 3 0 4
5. Determine (draw) the image of the unit circle under the linear transformation
"
# that is
1 1
represented (with respect to the standard basis) by the matrix L =
.
0 1
"
#
a b
6. Let A =
denote a general (2 × 2)-matrix for which a 6= 0 and ad − bc 6= 0.
c d
Show, by using row reduction, the computation of the inverse matrix of A.
Math 237
Carter
Test 2 Spring 2009
General Instructions: Do not write on this test. Write your name on only the outside
of your blue books. Do all your work inside your blue books. Write neat complete answers
to the questions below. Show your work.
To make lemonade or limeade, take between 4 and 6 of the fruits, cut each along the
equator. Squeeze using either an electric or hand reamer. Pour the juice into the bottom
of a 2 quart pitcher. Now add granulated sugar quickly and until the level of the sugar is
roughly at the level of the juice. If possible add the fruit pulp back to the mixture. Adjust
number of lemons, sugar, and water to taste. Fill the pitcher with cold water and ice, and
serve over ice.
1. Consider the matrix

1 1 1 1 1

 2 2 1 1 1
A=
 1 0 2 3 2

4 3 2 1 0



.


Give bases for the row space, the column space, and the null space.
6
2. Consider the vectors X = [1, 0, −1]t and Y = [0, −1, 1]t . (a) Determine an equation
of a plane through the origin that contains these two vectors. (b) Let Z = [0, 0, 1]t ;
determine an equation of a plane that contains {Z + aX + bY : For all a, b ∈ R}.
3. Find a basis for the null space of the matrix:


0 1 0 2 4



A=
0
0
1
2
4


0 0 0 0 1
4. Without performing any computation, explain why {[1, 2, 4], [2, 3, 5], [5, 8, 6], [−10, 4, 7]}
is not linearly independent.
5. State three different characterizations of a non-singular (n × n)-matrix. Give an argument to prove the equivalence of any two of these.
6. Describe the procedure to determine a basis for the row space of a matrix.
7. Will a non-zero vector in the null space of a matrix ever be found in the row space?
8. Determine the row space, column space, and null

2 1 1

A=
 1 3 2
5 0 1
9. Consider the matrix

space of the matrix:




2 4 −3 5


1 
.
4 6 −1 7

A=
 1 1
1
(a) What is the rank of A? (b) Show that there is a vector B ∈ M (3, 1) for which
there is no solution to the equation AX = B. (c) Suppose that AX = B 0 (Different
right hand side!) has a solution. Give a description of the solution space.
10. An (n × n)-matrix, C, is said to be nilpotent if there is an integer n such that C n = 0.
Prove that the matrix

0 a b c



 0 0 d e 

A=
 0 0 0 f 


0 0 0 0
is nilpotent.
7
11. Compute the eigenvalues and corresponding eigenvectors for the matrix
"
#
13 2
A=
4 11
12. Let P be an arbitrary vector in M (n, 1). Show that P is an eigenvector of the matrix
MP = I −
2
PPt
P tP
with eigenvalue −1.
Math 237
Carter
Test 1 Fall 2010
General Instructions: Do not write on this test. Write your name on only the outside of
your blue books. Do all your work inside your blue books. Write neat complete answers to
the questions below. Remember to use your turn signals when changing lanes. Show your
work.
1. Define the italicized terms (5 points each):
(a) The set {A1 , A2 , . . . , Ak } is linearly independent
(b) The set {A1 , A2 , . . . , Ak } spans a vector space V
(c) The null space of an (m × n)-matrix
(d) The span of a set of vectors {A1 , A2 , . . . , Ak }
2. (10 points) The reduced row echelon form of the matrix A that is associated to the
homogeneous system of equations
5x + 4y + 3z + 6w = 0
x
+ 3y + 2z + 2w = 0
3x − 2y −
is

1 0

 0 1

0 0
z
+ 2w = 0
1
11
7
11
10
11
4
11
0
0
(a) Determine the solution set.
(b) Give a spanning set for the column space.
8




3. Solve the system of equations (10 points)
x
+ 2y
=
3
4x + 8y + 2z = 14
x
+ 2y +
z
=
4
4. (10 points) Compute the matrix product:
"
# "
#
1/5 4/5
5 −4 20
·
0
1
0 1 0
5. (a) (5 points) Write the augmented matrix for the system of equations:
x
+ y = 1
2x − y = 2
(b) (15 points) By successively performing
"
# the corresponding row operations to the
1 0
(2 × 2)-identity matrix, I =
, solve the system of equations.
0 1
6. (10 points) Give an example of a linearly dependent set of 3 vectors in M (2, 1) such
that any two vectors taken from the set form a linearly independent set.
7. (10 points) Give an equation of the form Ax + By + Cz = 0 such that the vectors
X = [1, 2, 1]t and Y = [1, 0, 3]t satisfy the equation.
8. (10 points) Is the matrix


0 1 2 2 4



A= 0 0 1 2 4 

0 0 0 0 1
in reduced row echelon form? If not, then row reduce it. Indicate your steps clearly.
Math 237
Carter
Test 2 Fall 2010
General Instructions: Do not write on this test. Write your name on only the outside
of your blue books. Do all your work inside your blue books. Write neat complete answers
to the questions below. Show your work. Yield to pedestrians and bicyclists. Good luck.
1. Let V denote a vector space. Fill in the blanks: (5 points each)
9
(a) The span of the set {v1 , v2 , . . . , vk } ⊂ V is the set
{v ∈ V :
}
(b) If the homogeneous equation
x1 v1 + x2 v2 + . . . xk vk = 0
has only the trivial solution x1 = x2 = . . . = xk = 0, then the set {v1 , v2 , . . . , vk } ⊂
V is said to be
.
of a matrix A ∈ M (m × n) is the span of the set of rows.
(c) The
(d) A basis for a vector space V is a set {v1 , v2 , . . . , vk } ⊂ V such that
i.
ii.
(e) If the vector space V has a basis {v1 , v2 , . . . , vk }, then the dimensions of V is
.
2. (10 points) Let A be a given (3 × 2)-matrix. Explain why there is an equation AX = B
which has no solution.
3. (10 points) Show that the set of symmetric matrices S = {A ∈ M (n, n) : A = At } is a
subspace of M (n, n). What is its dimension? Describe a basis.
4. (10 points) Determine the image of the unit square (the square with vertices (0, 0)t ,
(1, 0)t , (1, 1)t , (0, 1)t ) under the linear transformation
" # "
#
x
3x − y
L
=
.
y
x − 2y
5. (10 points) Determine a basis for the null space of the matrix
"
#
3 2 0 2
.
0 2 3 −2
6. (25 points) The reduced row echelon form of

−1 −2

A=
2
 1
2
4
10
the matrix

8 1 3

4 1 −1 

2 1 −3
is

1 2 0
1
3
1
6

A0 = 
 0 0 1
0 0 0 0
− 53
1
6
0


.

(a) Give a basis for the column space of A.
(b) Give a basis for the row space of A.
(c) Give a basis for the null space of A.
(d) Write the vector [1, 1, 1]t as a linear combination of the basis vectors of the column
space that you found above.
(e) The column space defines a plane in M (3, 1). Give an equation for this plane in
the form Ax + By + Cz = 0 where A, B, and C are constants.
7. (10 points) A matrix A ∈ M (4, 7) has rank 2. What is the dimension of the null
space? Give an example of such a matrix, and a vector B ∈ M (4, 1) such that there
is no solution, X, to the equation AX = B. You may choose A to be in reduced row
echelon form.
11