Proving an injective map exists from $V$ to $W \iff$ dim $V$≤ dim$W$ 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.
The map $T$ is defined by
$$
T(a_1v_1+a_2v_2+\dots+a_nv_n)=a_1w_1+a_2w_2+\dots+a_nw_n
$$
(you forgot the + signs). It is injective by the fundamental theorem, because the image is spanned by $\{w_1,\dots,w_n\}$, which is linearly independent. Thus the image has dimension $n$.
1) The statement is wrong because there are no restriction on map exept injectivity. So we can find injective map from $\mathbb R^2$ to  $\mathbb R$ because they are have the same cardinality.

2) The statement will be true if a linear injective mapping is implied