Linear Algebra, Theory And Applications, 2012a

2.1. MATRICES 39
Definition 2.1.2 Matrices which are n × 1 or 1 × n are especially called vectors and are
often denoted by a bold letter. Thus
⎛
⎜
x = ⎝
x 1
.
x n
⎞
⎟
⎠
is an n × 1 matrix also called a column vector while a 1 × n matrix of the form (x 1 ···x n )
is referred to as a row vector.
All the above is fine, but the real reason for considering matrices is that they can be
multiplied. This is where things quit being banal.
First consider the problem of multiplying an m × n matrix by an n × 1 column vector.
Consider the following example
It equals
( 1 2 3
4 5 6
( 1
7
4
) ( 2
+8
5
) ⎛ ⎝ 7 8
9
⎞
⎠ =?
) ( 3
+9
6
Thus it is what is called a linear combination of the columns. These will be discussed
more later. Motivated by this example, here is the definition of how to multiply an m × n
matrix by an n × 1 matrix. (vector)
Definition 2.1.3 Let A = A ij be an m × n matrix and let v be an n × 1 matrix,
⎛
⎜
v = ⎝
v 1
. ⎞
⎟
⎠ , A =(a 1 , ··· , a n )
v n
)
where a i is an m × 1 vector. Then Av, written as
⎛
( ) ⎜
a1 ··· a n ⎝
v 1
. .
⎞
⎟
⎠ ,
is the m × 1 column vector which equals the following linear combination of the columns.
n∑
v 1 a 1 + v 2 a 2 + ···+ v n a n ≡ v j a j (2.9)
If the j th column of A is
then (2.9) takes the form
⎛
A 11
A 21
v 1 ⎜ .
⎝ .
A m1
⎞ ⎛
⎟
⎠ + v 2 ⎜
⎝
⎛
⎜
⎝
A 1j
A 2j
.
A mj
A 12
A 22
.
.
A m2
⎞
⎟
⎠
v n
j=1
⎞ ⎛
⎟
⎠ + ···+ v n ⎜
⎝
A 1n
A 2n
.
.
A mn
⎞
⎟
⎠