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.