Page 49 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Matrices Defining Functions

The equation with A and b given defines a linear system to be solved for x. But we could also write this equation as

, where A and x are given. In this case, we want to compute y. If A is , then this is a function that associates with

every column vector x an column vector y, and we may view A as defining a rule that shows how a given x is

mapped into a corresponding y. This idea is discussed in more detail starting in Section 4.2.

EXAMPLE 10 A Function Using Matrices
Consider the following matrices.

The product is

so the effect of multiplying A by a column vector is to change the sign of the second entry of the column vector. For the
matrix

the product is

so the effect of multiplying B by a column vector is to interchange the first and second entries of the column vector, also
changing the sign of the first entry.

If we view the column vector x as locating a point in the plane, then the effect of A is to reflect the point about the

x-axis (Figure 1.3.1a) whereas the effect of B is to rotate the line segment from the origin to the point through a right angle

(Figure 1.3.1b).