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.)

Definitions.

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 ...

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