image of a linear transformation

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

 $\operatorname{Im}(T)=\{w\in W\mid w=T(v)\,\mbox{for some}\,v\in V\}=T(V).$

0.0.1 Properties

1. 1.

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

2. 2.

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

3. 3.

$\operatorname{Im}(T)$ is a vector subspace of $W$;

4. 4.

If $L\colon W\to U$ is a linear transformation, then $\operatorname{Im}(LT)=L(\operatorname{Im}(T))$;

Title image of a linear transformation ImageOfALinearTransformation 2013-03-22 13:48:32 2013-03-22 13:48:32 Koro (127) Koro (127) 8 Koro (127) Definition msc 15A04 RankNullityTheorem KernelOfALinearTransformation