Suppose $v_1, \dots v_n$ spans V and $T \in L(V, W)$. Prove that the list $Tv_1, \dots , Tv_n$ spans rangeT.
I said that if $v \in V = a_1v_1 + \dots + a_nv_n$
then $T(a_1v_1 + \dots + a_nv_n) = a_1w_1 + \dots + a_nw_n$ but how do I show this spans W if I don't know that T is surjective? Thanks!