2.1 Linear Transformations, Null Spaces, and Ranges

In this section, we consider a number of examples of linear transformations. Many of these transformations are studied in more detail in later sections. Recall that a function T with domain V and codomain W is denoted by T:VW. (See Appendix B.)


Let V and W be vector spaces over the same Geld F. We call a function T:VW a linear transformation from V to W if, for all x,yV and cF, we have

  1. T(x+y)=T(x)+T(y) and

  2. T(cx)=cT(x).

If the underlying field F is the field of rational numbers, then (a) implies (b) (see Exercise 38), but, in general (a) and (b) are logically independent. See Exercises 39 and 40.

We often simply call T linear. The reader should verify the following properties ...

Get Linear Algebra, 5th Edition now with O’Reilly online learning.

O’Reilly members experience live online training, plus books, videos, and digital content from 200+ publishers.