Linear Algebra, Theory And Applications, 2012a

40 MATRICES AND LINEAR TRANSFORMATIONS
Thus the i th entry of Av is ∑ n
j=1 A ijv j . Note that multiplication by an m × n matrix takes
an n × 1 matrix, and produces an m × 1 matrix (vector).
Here is another example.
Example 2.1.4 Compute
⎛
⎛
⎝ 1 2 1 3
⎞
0 2 1 −2 ⎠ ⎜
⎝
2 1 4 1
1
2
0
1
⎞
⎟
⎠ .
First of all, this is of the form (3 × 4) (4 × 1) and so the result should be a (3 × 1) .
Note how the inside numbers cancel. To get the entry in the second row and first and only
column, compute
4∑
a 2k v k = a 21 v 1 + a 22 v 2 + a 23 v 3 + a 24 v 4
k=1
You should do the rest of the problem and verify
⎛
⎛
⎝ 1 2 1 3 ⎞
0 2 1 −2 ⎠ ⎜
⎝
2 1 4 1
= 0× 1+2× 2+1× 0+(−2) × 1=2.
1
2
0
1
⎞
⎛
⎟
⎠ = ⎝ 8 2
5
With this done, the next task is to multiply an m × n matrix times an n × p matrix.
Before doing so, the following may be helpful.
these must match
(m × ̂n) (n × p )=m × p
If the two middle numbers don’t match, you can’t multiply the matrices!
Definition 2.1.5 Let A be an m × n matrix and let B be an n × p matrix. Then B is of
the form
B =(b 1 , ··· , b p )
where b k is an n × 1 matrix. Then an m × p matrix AB is defined as follows:
⎞
⎠ .
AB ≡ (Ab 1 , ··· ,Ab p ) (2.10)
where Ab k is an m × 1 matrix. Hence AB as just defined is an m × p matrix. For example,
Example 2.1.6 Multiply the following.
(
1 2 1
0 2 1
) ⎛ ⎝
1 2 0
0 3 1
−2 1 1
The first thing you need to check before doing anything else is whether it is possible to
do the multiplication. The first matrix is a 2×3 and the second matrix is a 3×3. Therefore,
⎞
⎠