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The
matrix-vector product
between an
m
×
n
matrix
A
and a column
n
-vector
x
is the
column
m
-vector
b
below:
⎡
⎤
⎡
⎣
⎤
a
1,1
a
1,2
...
a
1,
n
x
1
x
2
.
x
n−
1
x
n
⎣
⎦
⎦
a
2,1
a
2,2
...
a
2,
n
.
.
.
.
.
.
b
=
A
x
=
×
a
m
,1
a
n−
1,2
...
a
m−
1,
n
a
m
,1
a
n
,2
...
a
m
,
n
⎡
⎤
a
1,1
∗
x
1
+
a
1,2
∗
x
2
+
···
+
a
1,
n
∗
x
n
⎣
⎦
a
2,1
∗
x
1
+
a
2,2
∗
x
2
+
···
+
a
2,
n
∗
x
n
.
.
.
.
.
.
=
a
m−
1,1
∗
x
1
+
a
m−
1,2
∗
x
2
+
···
+
a
m−
1,
n
∗
x
n
a
m
,1
∗
x
1
+
a
m
,2
∗
x
2
+
···
+
a
m
,
n
∗
x
n
Thus,
b
j
is defined as follows for 1
≤
j
≤
n
:
b
j
=
a
i
,1
∗
x
1
+
a
i
,2
∗
x
2
+
···
+
a
i
,
m
∗
x
m
The matrix-vector product between a row
m
-vector
x
and an
m
×
n
matrix
A
is the row
n
-vector
b
below:
b
=[
b
i
]=
x
A
≤
i
≤
nb
i
satisfies
where for 1
b
i
=
x
1
∗
a
1,
i
+
x
2
∗
a
2,
i
+
···
+
x
m
∗
a
m
,
i
The special case of a matrix-vector product between a row
n
-vector,
x
, and a column
n
vector,
y
, denoted
x
·
y
and defined below, is called the
inner product
of the two vectors:
n
x
·
y
=
x
i
∗
y
i
i
=
1
If the entries of the
n
×
n
matrix
A
and the column
n
-vectors
x
and
b
shown below are
drawn from a ring
and
A
and
b
are given, then the following matrix equation defines a
linear system
of
n
equations in the
n
unknowns
x
:
R
A
x
=
b
An example of a linear system of four equations in four unknowns is
∗
x
1
+
∗
x
2
+
∗
x
3
+
∗
x
4
=
1
2
3
4
17
∗
x
1
+
∗
x
2
+
∗
x
3
+
∗
x
4
=
5
6
7
8
18
∗
x
1
+
∗
x
2
+
∗
x
3
+
∗
x
4
=
9
10
11
12
19
13
∗
x
1
+
14
∗
x
2
+
15
∗
x
3
+
16
∗
x
4
=
20
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