Linear Algebra with Applications, 10th Edition

4.1 Definition and Examples

In the study of vector spaces, the most important types of mappings are linear transformations.

Definition

A mapping L from a vector space V into a vector space W is said to be a linear transformation if

$$L(\alpha {\mathbf{\text{v}}}_1+\beta {\mathbf{\text{v}}}_2)=\alpha L\left({\mathbf{\text{v}}}_1\right)+\beta L\left({\mathbf{\text{v}}}_2\right)$$ (1)

for all ${\mathbf{\text{v}}}_1,{\mathbf{\text{v}}}_2\in V$ and for all scalars $\alpha$ and $\beta$.

If L is a linear transformation mapping a vector space V into a vector space W, then it follows from (1) that

$$\begin{array}{cc}L({\mathbf{\text{v}}}_1+{\mathbf{\text{v}}}_2)=L\left({\mathbf{\text{v}}}_1\right)+L\left({\mathbf{\text{v}}}_2\right)& (\alpha =\beta =1)\end{array}$$ (2)

and

$$\begin{array}{cc}L\left(\alpha \mathbf{\text{v}}\right)=\alpha L\left(\mathbf{\text{v}}\right)& (\mathbf{\text{v}}={\mathbf{\text{v}}}_1,\beta =0)\end{array}$$ (3)

Conversely, if L satisfies (2) and (3), then

$$\begin{array}{cc}\hfill L(\alpha {\mathbf{\text{v}}}_1+\beta {\mathbf{\text{v}}}_2)& =L(\alpha {\mathbf{\text{v}}}_1)+L(\beta {\mathbf{\text{v}}}_2)\hfill \\ =\alpha L\left({\mathbf{\text{v}}}_1\right)+\beta L\left({\mathbf{\text{v}}}_2\right)\hfill \end{array}$$

Thus, L is a linear transformation if and only if L satisfies (2) and (3).

Notation

A mapping L from a vector space V into a vector space W will be denoted