# 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*

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

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

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 ...

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