image of a linear transformation

DefinitionLet $T:V\to W$ be a linear transformation. Then the image of$T$ is the set

$$\mathrm{Im}(T)=\{w\in W\mid w=T(v)\text{for some}v\in V\}=T(V).$$

0.0.1 Properties

1. The dimension of $\mathrm{Im}(T)$ is called the rank of $T$;

2. $T$ is a surjection, if and only if $\mathrm{Im}(T)=W$;

3. $\mathrm{Im}(T)$ is a vector subspace of $W$;

4. If $L:W\to U$ is a linear transformation, then $\mathrm{Im}(LT)=L(\mathrm{Im}(T))$;