Basis in a Linear Map

Question

Suppose we have a linear map $T\in\mathcal{L}(V,W)$ and $(v_1,...,v_n)$ is the basis of $V$. Does that necessary mean $(Tv_1,\cdots,Tv_n)$ is the basis for $W$?

Answer 1

No. Suppose that $T$ is the null map. Then your set is just the null vector and therefore it is not a basis.

Answer 2

No, as others have already pointed out. However, if $T:V\rightarrow W$ is injective and $\{v_i\}$ are linearly independent, then $\{T(v_i)\}$ are also linearly independent. If $T$ is also surjective (hence invertible) and $\{v_i\} $ is a basis of $V$ then $\{T(v_i)\}$ is a basis of $W$.

You write the basis of a vector space, but a generic vector space has no preferred basis, and there always are many possible bases (a very simple way to generate new bases from a given one is to change the sign of some basis elements, but you can do much more than that). Therefore, it would be better to write (and think) of a basis.