In my Linear Algebra and Geometry textbook, it defines the image of a linear transformation $T$ as:
$\operatorname{Im}\, (T) := \{\; w \in W : \; w=Tv \;\;\text{ for some } v \in V \} $
As far as I can see, this is just the same as:
$\operatorname{Im} \, (T) := \{ \;Tv \in W : \;v \in V\}$
Is there any difference in these definitions?
If not, why is the first one used?