Page 180 - A Course in Linear Algebra with Applications

164 Chapter Six: Linear Transformationns

Conversely, suppose that we start with an m x n matrix
n m
A over F; then we can define a function T : F —> F by the
rule T(X) = AX. The laws of matrix algebra guarantee that
T is a linear transformation; for by 1.2.1

A(Xi + X 2) = AX X + AX 2 and A(cX) = c(AX).

We have now established a fundamental connection between
matrices and linear transformations.

Theorem 6.2.2
(i) Let T : F n —> F m be a linear transformation. Then
n
T{X) = AX for all X in F where A is the m x n matrix
whose columns are the images under T of the standard basis
n
vectors of F .
(ii) Conversely, if A is any mx n matrix over the field F, the
function T : F n -> F m defined by T(X) = AX is a linear
transformation.

Example 6.2.6
Define T : R 3 -»• R 2 by the rule

One quickly checks that T is a linear transformation. The
images under T of the standard basis vectors Ei, E 2, E3 are

respectively. It follows that T is represented by the matrix

A =
[ 0 - 1 3 ) '