Page 97 - Matrix Analysis & Applied Linear Algebra

3.3 Linearity 91
Example 3.3.2

Consider a linear system
a 11 x 1 + a 12 x 2 + ··· + a 1n x n = u 1 ,
a 21 x 1 + a 22 x 2 + ··· + a 2n x n = u 2 ,
.
.
.
a m1 x 1 + a m2 x 2 + ··· + a mn x n = u m ,
   
x 1 u 1
 x 2   u 2 
m
to be a function u = f(x) that maps x =  .  ∈ n to u =  .  ∈ .
. .
 .   . 
x n u m
Problem: Show that u = f(x) is linear.
Solution: Let A =[a ij ] be the matrix of coefficients, and write
 
αx 1 + y 1
n n

 αx 2 + y 2 
f(αx + y)= f  .  = (αx j + y j )A ∗j = (αx j A ∗j + y j A ∗j )
.
 . 
j=1 j=1
αx n + y n
n n n n

= αx j A ∗j + y j A ∗j = α x j A ∗j + y j A ∗j
j=1 j=1 j=1 j=1
= αf(x)+ f(y).

According to (3.3.3), the function f is linear.

The following terminology will be used from now on.

Linear Combinations

For scalars α j and matrices X j , the expression
n

α 1 X 1 + α 2 X 2 + ··· + α n X n = α j X j
j=1

is called a linear combination of the X j ’s.