Page 586 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Figure 8.1.4

translates each point along a line parallel to through a distance .

Finding Linear Transformations from Images of Basis Vectors

Theorem 4.3.3 shows that if T is a matrix transformation, then the standard matrix for T can be obtained from the images of

the standard basis vectors. Stated another way, a matrix transformation is completely determined by its images of the

standard basis vectors. This is a special case of a more general result: If is a linear transformation, and if

is any basis for V, then the image of any vector v in V can be calculated from the images

of the basis vectors. This can be done by first expressing v as a linear combination of the basis vectors, say
and then using Formula 1 to write

In words, a linear transformation is completely determined by the images of any set of basis vectors.

EXAMPLE 14 Computing with Images of Basis Vectors

Consider the basis for , where , , and . Let be the

linear transformation such that

Find a formula for ; then use this formula to compute .

Solution as a linear combination of , , and . If we write

We first express

then on equating corresponding components, we obtain

which yields , , , so

Thus