existence of a linear map Let $V$ and $W$ be finite dimensional spaces. Given a positive integer $m$ and vectors $v_1, .., v_m \in V$ and $w_1, .., w_m \in W$. We assume that for every linear combination  $\sum_{i=1}^m a_i v_i = 0$, we have $\sum_{i=1}^m a_i w_i = 0$. I want to show that there exists a linear map $T : V \to W$ satisfying $T v_j = w_j$. 
I say the following : Denote $V' = span(v_1,...,v_m)$ and $W' = span(w_1,...,w_m)$. V' has a basis $\{e_1, ..., e_ p \}$ and W' has a basis $\{f_1, ..., f_ q \}$ with $p,q \leq m$.
There exist some number $\alpha_{ij}$ and $\beta_{ik}$ such that 
$v_i = \sum_{j=1}^p \alpha_{ij} e_j$ and $w_i = \sum_{k=1}^q \beta_{ik} f_k$.
Now I want to use the hypothesis, to justify that we can find $T$
 such that 
$$
\sum_{j=1}^p \alpha_{ij} Te_j = \sum_{k=1}^q \beta_{ik} f_k
$$
which boils down into finding some numbers $T_{jk}$ such that 
$$
\sum_{j=1}^p \alpha_{ij} \left (\sum_{k=1}^q Tjk f_k \right ) = \sum_{k=1}^q \beta_{ik} f_k.
$$
How the hypothesis helps to justify that the numbers $T_{jk}$ exist?
Thanks.

It is simpler if you take the basis $e_1,\ldots,e_p$ to be a subset of $v_1,\ldots,v_m$. Without loss of generality, you can take $e_i=v_i,\quad i=1,\ldots, p.$ Let $T:V'\to W$ be the unique linear transformation satisfying $T(e_i)=w_i,\quad i=1,\ldots,p$. It follows from the hypothesis that $T$ is the required transformation.
