How do I prove that a spanning set when transformed spans the range of a linear map T?

Question

I started out with the following reasoning: Consider a spanning set $v \in V: v_1, ..., v_n$. By definition every element in $V$ is some linear combination of this list. By linearity, $T(\alpha v_1 + ... + \alpha v_n)$ = $\alpha T(v_1) + ... +\alpha T(v_n)$, which belongs to the Range of T.

I'm not sure how to finish this and put everything together.

Answer 1

Let $w \in \text{Ran}{(T)}\implies w = T(u), u \in V\implies u = a_1v_1+a_2v_2+\cdots+a_nv_n\implies w = T(u) = a_1T(v_1)+a_2T(v_2)+\cdots +a_nT(v_n)$. This implies the claim.