A First Course in Linear Algebra, 2017a

60 Matrices
on the left by a vector on the right, the numbers making up the vector are just the scalars to be used in the
linear combination of the columns as illustrated above.
Here is the formal definition of how to multiply an m × n matrix by an n × 1 column vector.
Definition 2.13: Multiplication of Vector by Matrix
Let A = [ a ij
]
be an m × n matrix and let X be an n × 1 matrix given by
A =[A 1 ···A n ],X =
⎡
⎢
⎣
⎤
x 1...
⎥
⎦
x n
Then the product AX is the m × 1 column vector which equals the following linear combination of
the columns of A:
n
x 1 A 1 + x 2 A 2 + ···+ x n A n = ∑ x j A j
j=1
If we write the columns of A in terms of their entries, they are of the form
⎡ ⎤
a 1 j
a 2 j
A j = ⎢ ⎥
⎣ . ⎦
a mj
Then, we can write the product AX as
⎡ ⎤ ⎡
a 11
a 21
AX = x 1 ⎢ ⎥
⎣ . ⎦ + x 2 ⎢
⎣
a m1
⎤
a 12
a 22
⎥
.
a m2
⎡
⎦ + ···+ x n ⎢
⎣
⎤
a 1n
a 2n
⎥
. ⎦
a mn
Note that multiplication of an m × n matrix and an n × 1 vector produces an m × 1 vector.
Here is an example.
Example 2.14: A Vector Multiplied by a Matrix
Compute the product AX for
⎡
A = ⎣
1 2 1 3
0 2 1 −2
2 1 4 1
⎤
⎦,X =
⎡
⎢
⎣
1
2
0
1
⎤
⎥
⎦
Solution. We will use Definition 2.13 to compute the product. Therefore, we compute the product AX as