Linear Algebra, 5th Edition

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 $\text{T}:\text{V}\to \text{W}$. (See Appendix B.)

Definitions.

Let V and W be vector spaces over the same Geld F. We call a function $\text{T}:\text{V}\to \text{W}$ a linear transformation from V to W if, for all $x,y\in \text{V}$ and $c\in F$, we have

1. $\text{T}(x+y)=\text{T}(x)+\text{T}(y)$ and

2. $\text{T}(cx)=c\text{T}(x)$.

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 ...