Page 585 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

and, more generally, if , ,…, are vectors in V and , , …, are scalars, then

(1)

Formula 1 is sometimes described by saying that linear transformations preserve linear combinations.
The following theorem lists three basic properties that are common to all linear transformations.

THEOREM 8.1.1

If is a linear transformation, then
(a)
(b) for all in V
(c) for all and in V

Proof Let v be any vector in V. Since , we have

which proves (a). Also, ; thus
which proves (b). Finally,

which proves (c).

In words, part (a) of the preceding theorem states that a linear transformation maps 0 to 0. This property is useful for
identifying transformations that are not linear. For example, if is a fixed nonzero vector in , then the transformation

has the geometric effect of translating each point x in a direction parallel to through a distance of (Figure 8.1.4).

This cannot be a linear transformation, since , so T does not map 0 to 0.