Theorem 4.2
Let
A
be an
m
×
n
matrix
A
=
a
11
a
12
· · ·
a
1
n
a
21
a
22
· · ·
a
2
n
.
.
.
.
.
.
.
.
.
.
.
.
a
m
1
a
m
2
· · ·
a
mn
.
and denote the columns of
A
by the column vectors
c
1
,
c
2
, . . . ,
c
n
, so that
c
i
=
a
1
i
a
2
i
.
.
.
a
mi
,
1
≤
i
≤
n.
Then if
x
= (
x
1
, x
2
, . . . , x
n
)
T
is any vector in
R
n
,
A
x
=
x
1
c
1
+
x
2
c
2
+
· · ·
x
n
c
n
.
This theorem states that the matrix product
A
x
, which is a vector in
R
m
, can be
expressed as a linear combination of the column vectors of
A
.
Activity 4.3
Prove this theorem; derive expressions for both the lefthand side and
the righthand side of the equality as a single
m
×
1 vector and compare the
components to prove the equality.
Read
section 1.8 of the text AH, working through the activities there. You will
find the solution of the last activity in the text.
4.2
Developing geometric insight –
R
2
and
R
3
Vectors have a broader use beyond that of being special types of matrices. It is likely
that you have some previous knowledge of vectors; for example, in describing the
displacement of an object from one point to another in
R
2
or in
R
3
. Before we continue
our study of linear algebra it is important to consolidate this background, for it
provides valuable geometric insight into the definitions and uses of vectors in higher
dimensions. Parts of this section may be review for you.
53
4
4. Vectors
4.2.1
Vectors in
R
2
The set
R
can be represented as points along a horizontal line, called a
realnumber line
.
In order to represent pairs of real numbers, (
a
1
, a
2
), we use a
Cartesian plane
, a plane
with both a horizontal axis and a vertical axis, each axis being a copy of the
realnumber line, and we mark
A
= (
a
1
, a
2
) as a
point
in this plane. We associate this
point with the vector
a
= (
a
1
, a
2
)
T
, as representing a
displacement
from the origin (the
point (0
,
0)) to the point
A
. In this context,
a
is the
position vector
of the point
A
.
This displacement is illustrated by an arrow, or directed line segment, with initial point
at the origin and terminal point at
A
.

x
6
y
(0
,
0)
>
r
(
a
1
, a
2
)
a
2
a
1
a
position vector,
a
Even if a displacement does not begin at the origin, two displacements of the same
length and the same direction are considered to be equal. So, for example, the two
arrows below represent the same vector,
v
= (1
,
2)
T
.

x
6
y
(0
,
0)
displacement vectors,
v
If an object is displaced from a point, say
O
, the origin, to a point
P
by the
displacement
p
, and then displaced from
P
to
Q
, by the displacement
v
, then the total
displacement is given by the vector from
O
to
Q
, which is the position vector
q
. So we
would expect vectors to satisfy
q
=
p
+
v
, both geometrically (in the sense of a
displacement) and algebraically (by the definition of vector addition). This is certainly
true in general, as illustrated below.
54
4
4.2. Developing geometric insight –
R
2
and
R
3

6
(0
,
0)
:
>
*
p
2
p
1
q
2
q
1
p
v
q
If
v
= (
v
1
, v
2
)
T
, then
q
1
=
p
1
+
v
1
and
q
2
=
v
2
+
p
2
.