Page 26 - 35Linear Algebra

26 What is Linear Algebra?

the new function obtained by plugging the outputs if f into g is called g ◦ f,

g ◦ f : U −→ W

where
(g ◦ f)(u) = g(f(u)) .
This is called the composition of functions. Matrix multiplication is the tool
required for computing the composition of linear functions.

1.4.2 The Matrix Detour

Linear algebra is about linear functions, not matrices. The following presen-
tation is meant to get you thinking about this idea constantly throughout
the course.

Matrices only get involved in linear algebra when certain
notational choices are made.

To exemplify, lets look at the derivative operator again.

Example 9 of how matrices come into linear algebra.
Consider the equation

d
+ 2 f = x + 1
dx
where f is unknown (the place where solutions should go) and the linear differential
2
operator d +2 is understood to take in quadratic functions (of the form ax +bx+c)
dx
and give out other quadratic functions.
Let’s simplify the way we denote the quadratic functions; we will

 
a
2
denote ax + bx + c as   .
b
c
B
The subscript B serves to remind us of our particular notational convention; we will
compare to another notational convention later. With the convention B we can say

 
a

d d 2
+ 2   = + 2 (ax + bx + c)
b
dx dx
c
B
26