Math 1b — Matrix Multiplication
If
A
has rows
a
i
and
B
has columns
b
j
,then
AB
has, by deFnition,
a
i
b
j
as the entry
in row
i
and column
j
. It is the matrix of dot products.
Here are some simple properties and facts about matrix multiplication. These rules
follow directly from the deFnition of matrix multiplication.
Small examples can help
understanding.
1. A (row) vector times a matrix is a linear combination of the rows of that matrix
(and the coeﬃcients are the entries of the vector):
(
c
1
c
2
... c
`
)
⎛
⎜
⎜
⎝
—
a
1
—
—
a
2
—
.
.
.
—
a
`
—
⎞
⎟
⎟
⎠
=
c
1
a
1
+
c
2
a
2
+
...
+
c
`
a
`
.
2. A matrix times a (column) vector is a linear combination of the columns of that
matrix (and the coeﬃcients are the entries of the vector).
⎛
⎝


b
1
b
2
b
k

⎞
⎠
⎛
⎜
⎜
⎝
c
1
c
2
.
.
.
c
k
⎞
⎟
⎟
⎠
=
c
1
⎛
⎝

b
1

⎞
⎠
+
c
2
⎛
⎝

b
2

⎞
⎠
+
+
c
k
⎛
⎝

b
k

⎞
⎠
.
3. The rows of the matrix product
AB
are the rows of
A
times
B
.
⎛
⎜
⎜
⎝
—
a
1
—
—
a
2
—
.
.
.
—
a
`
—
⎞
⎟
⎟
⎠
B
=
⎛
⎜
⎜
⎝
—
a
1
B
—
—
a
2
B
—
.
.
.
—
a
`
B
—
⎞
⎟
⎟
⎠
.
4. The columns of
AB
are
A
times the columns of
B
.
A
⎛
⎝

b
1
b
2
b
k

⎞
⎠
=
⎛
⎝

A
b
1
A
b
2
... A
b
k

⎞
⎠
.
It is clear from this that
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 Winter '09
 Bopanna
 Linear Algebra, Multiplication, Vector Space, Dot Product, Ring

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