Page 282 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Figure 4.2.1

EXAMPLE 3 Zero Transformation from to

If 0 is the zero matrix and 0 is the zero vector in , then for every vector x in ,

so multiplication by zero maps every vector in into the zero vector in . We call the zero transformation from to
. Sometimes the zero transformation is denoted by 0. Although this is the same notation used for the zero matrix, the appropriate
interpretation will usually be clear from the context.

EXAMPLE 4 Identity Operator on

If I is the identity matrix, then for every vector x in ,

so multiplication by I maps every vector in into itself. We call the identity operator on . Sometimes the identity
operator is denoted by I. Although this is the same notation used for the identity matrix, the appropriate interpretation will
usually be clear from the context.

Among the most important linear operators on and are those that produce reflections, projections, and rotations. We shall
now discuss such operators.

Reflection Operators