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8.3 In Section 4.3 we discussed properties of one-to-one linear transformations
from to . In this section we shall extend those ideas to more general
INVERSE LINEAR kinds of linear transformations.
TRANSFORMATIONS
Recall from Section 4.3 that a linear transformation from to is called one-to-one if it maps distinct vectors in into
distinct vectors in . The following definition generalizes that idea.
DEFINITION is said to be one-to-one if T maps distinct vectors in V into distinct vectors in W.
A linear transformation
EXAMPLE 1 A One-to-One Linear Transformation
Recall from Theorem 4.3.1 that if A is an matrix and is multiplication by A, then is one-to-one if
and only if A is an invertible matrix.
EXAMPLE 2 A One-to-One Linear Transformation
Let be the linear transformation
discussed in Example 8 of Section 8.1. If
are distinct polynomials, then they differ in at least one coefficient. Thus,
also differ in at least one coefficient. Thus T is one-to-one, since it maps distinct polynomials and into distinct
polynomials and .
EXAMPLE 3 A Transformation That Is Not One-to-One
Calculus Required
