A set of vectors span some space if the vectors are linearly independent.
The span doesn't need to span the dimension.
If we are working in , our span could be 2 vectors in the xy plane as long as they aren't linearly dependent.
A linear transformation in V maps the a vector v in V to W following some predefined rule.
L(x) = 3x.
All the vectors from V to W are 3 times longer in W.