Answer
See below
Work Step by Step
We are given $\{w_1,w_2...,w_m\}$ spans $W \rightarrow w \in W$ (1)
Let $\alpha_1,\alpha_2,...,\alpha_m$ be scarlars such as:
$$\alpha_1v_1+\alpha_2v_2+....+a_mv_m=0 \\
\rightarrow \alpha_1=\alpha_2=...=\alpha_m=0$$
Let:
$$w=\alpha_1 T(v_1),\alpha_2T(v_2),...,\alpha_mT(v_m) \\
w=T(\alpha_1(v_1)+\alpha_2(v_2)+...+\alpha_mv(v_m)) $$
Since $V$ is a vector space and $v_1,v_2....,v_m \in V \\
\rightarrow \alpha_1 v_1+\alpha_2 v_2+...+\alpha _mv_m \in V\\
\rightarrow v \in V$ (2)
From(1) and (2), we have $T(v)=w $
Hence $T$ is onto.
