Prove that if $W_1\subseteq V$ finite-dimensional, then there is $W_2\subseteq V$ such that $V=W_1\oplus W_2$

If you already know that you can compete a basis of a subspace to a basis for the whole space then you are practically done.

Hint: Note that $\alpha$ is a basis for $V$ (this should give you $W_{1}+W_{2}=V$, why ?) and that the $w_{i}$ are linearly independent of the $u_{i}$ (this should show that $W_{1}\cap W_{2}=\{0\}$, why ?)

Note: The way I see it, there is no use for $\beta$ or of the $v_{i}$ in the proof