Page 281 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

(6)

so the standard matrix for T is

The image of a point can be computed directly from the defining equations 5 or from 6 by matrix multiplication.
For example, if , then substituting in 5 yields

(verify) or alternatively from 6,

Some Notational Matters

If is multiplication by A, and if it is important to emphasize that A is the standard matrix for T, we shall denote the

linear transformation by . Thus

(7)

It is understood in this equation that the vector x in is expressed as a column matrix.

Sometimes it is awkward to introduce a new letter to denote the standard matrix for a linear transformation . In

such cases we will denote the standard matrix for T by the symbol .With this notation, equation 7 would take the form

(8)

Occasionally, the two notations for a standard matrix will be mixed, in which case we have the relationship

(9)

Remark Amidst all of this notation, it is important to keep in mind that we have established a correspondence between

matrices and linear transformations from to : To each matrix A there corresponds a linear transformation

(multiplication by A), and to each linear transformation , there corresponds an matrix (the standard

matrix for T).

Geometry of Linear Transformations

Depending on whether n-tuples are regarded as points or vectors, the geometric effect of an operator is to
transform each point (or vector) in into some new point (or vector) (Figure 4.2.1).