Page 168 - A Course in Linear Algebra with Applications

Chapter Six

LINEAR TRANSFORMATIONS

A linear transformation is a function between two vector
spaces which relates the structures of the spaces. Linear trans-
formations include operations as diverse as multiplication of
column vectors by matrices and differentiation of functions
of a real variable. Despite their diversity, linear transforma-
tions have many common properties which can be exploited
in different contexts. This is a good reason for studying linear
transformations and indeed much else in linear algebra.
In order to establish notation and basic ideas, we begin
with a brief discussion of functions defined on arbitrary sets.
Readers who are familiar with this elementary material may
wish to skip 6.1.

6.1 Functions Denned on Sets

If X and Y are two non-empty sets, a function or mapping
from X to Y,
F :X -> y,

is a rule that assigns to each element x o f l a unique element
F(x) of Y, called the image of x under F. The sets X and
Y are called the domain and codomain of the function F re-
spectively. The set of all images of elements of X is called the
image of the function F; it is written

Im(F).

Examples of functions abound; the most familiar are quite
likely the functions that arise in calculus, namely functions
whose domain and codomain are subsets of the set of real
numbers R. An example of a function which has the flavor

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