I'm reading Axler's Linear Alg Done Right and working through problems. For this one, I proved the forward direction already using Fundamental Theorem of Linear Maps.
And for the backwards direction I assumed dim $V$≤ dim $W$. Let $v_1,\ldots, v_n$ be a basis for $V$ and $w_1,\ldots, w_n$ be a basis for $W$. Then $T(a_1v_1 \ldots a_nv_n)=(a_1w_1\ldots a_mw_m)$.
I know I first I have to prove this is indeed a linear map, which I can do. Then I prove this is injective. I know an injective map null($T$)=$0$.
I'm stuck on proving this map is injective. Any help/guidance is appreciated.