image of a linear transformation
Definition Let $T:V\to W$ 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 $\mathrm{Im}(LT)=L(\mathrm{Im}(T))$;
Title  image of a linear transformation 

Canonical name  ImageOfALinearTransformation 
Date of creation  20130322 13:48:32 
Last modified on  20130322 13:48:32 
Owner  Koro (127) 
Last modified by  Koro (127) 
Numerical id  8 
Author  Koro (127) 
Entry type  Definition 
Classification  msc 15A04 
Related topic  RankNullityTheorem 
Related topic  KernelOfALinearTransformation 