Definition Let $T:V\to W$ be a linear transformation. Then the image of $T$ is the set $$ \operatorname{Im} (T) = \{ w\in W \mid w=T(v) \,\mbox{for some}\, v\in V\} = T(V).$$

Properties

1. The dimension of $\operatorname{Im}(T)$ is called the rank of $T$ ;
2. $T$ is a surjection, if and only if $\operatorname{Im}(T)=W$ ;
3. $\operatorname{Im}(T)$ is a vector subspace of $W$ ;
4. If $L\colon W\to U$ is a linear transformation, then $\operatorname{Im}(LT) =L(\operatorname{Im}(T))$ ;