Test 2
July 13, 2015
Ma 237-101 (CRN 30074)
Carter
General Instructions. Write your name on only the outside of your blue book. Do all of
your work inside your blue book. Do not write your answers on this test paper. Write neat,
complete solutions to each of the questions below. Please insert the test into your blue book
as you leave. There are 110 points. There are problems on the front and back. Instead of
oil try frying your eggs in butter. Melt the butter in a medium warm skillet while toasting
your bread. Sop up excess butter with bread before cooking the eggs.
1. Consider the (m × n)-matrix
 
 1
a1 a12 . . . a1n
 a2 a2 . . . a 2  
n 
2

 1
A= .
..
..  = 
.
 .
. ... .  
m
a1 am
. . . am
2
n
a1
a2
..
.

 
 = a1 a2 . . . an .

am
Define the following terms (5 points each):
(a) The Row Space of a matrix A.
(b) The Column Space of a matrix A.
(c) The Null Space of a matrix A.
2. State the rank/nullity theorem (5 points).
3. (5 points) One of the four statements below is not equivalent to the other three. Identify
the incorrect statement, and re-evaluate your answers in problem 1. Consider the
(n × n)-matrix
 1 1
  1 
a1 a2 . . . a1n
a
 a2 a2 . . . a2   a2  n 
 1 2


A= .
=  .  = a1 a2 . . . an .

.
.
.. . . . ..   .. 
 ..
an1 an2 . . . ann
an
(a) A represents a one-to-one (injective) linear transformation Rn ← Rn .
(b) A represents an onto (surjective) linear transformation Rn ← Rn .
(c) The row space of A is {a1 , a2 , . . . , an }.
(d) det (A) 6= 0.
1
4. (10 points) Consider the linear transformation T (x) = Ax for the given matrix A.
Determine if T is one-to-one (injective) and determine if T is onto (surjective).
1
4 −2
A=
.
3 −1
0
5. Let
T
x
y
x
y
=
2x + 3y
x − 2y
x − 2y
y
,
and
U
=
.
Find the matrices A, B, C, and D that represent the compositions indicated (5 points
each).
(a)
T (U (x)) = Ax,
(b)
U (T (x)) = Ax,
(c)
U (U (x)) = Ax,
(d)
T (T (x)) = Ax.
6. (15 points) Let
A=
3 −1
−5 2
.
Show that
A2 − 5A + I = 0
where I denotes the (2 × 2)-identity matrix.
7. (15 points) Use an augmented matrix
inverse of the matrix

1

 0

 0
0
and elementary row operations to compute the

2
1 −4

1 −2
2 

0
1
1 
0
0
1
2
8. (10 points) Find the steady state vectors ({x ∈ R2 : Ax = x}) for the matrix
A=
9. Let



A=

1
3
1
1
0.8 0.5
0.2 0.5
.
2 1
0
2
2
6 3
0
2
2
2 −3 4 −1 −3
2 3 −2 1 −3
Given that the reduced row echelon form

1 2

 0 0
A0 = 
 0 0
0 0
of A is
0 1 0 0
1 −1 0 0
0 0 1 0
0 0 0 1
(a) (5 points) Find a basis for the column space of A.
(b) (5 points) Determine the kernel (null space) of A.
(c) (5 points) Give a basis for the row space of A.
3









