Given a vector space $V$ and its dual $V^*$, we say $\alpha \in T^r(V^*)$ is an elementary tensor if there exist $\tilde{v}^1 , \dots, \tilde{v}^r \in V^*$ such that:
$$\alpha = \tilde{v}^1 \wedge \dots \wedge \tilde{v}^r .$$
If $\alpha \neq 0$, we can show that $W_\alpha = \{x \in V : \tilde{v}^j(x) = 0, j = 1, \dots , r \}$ is a subspace of $V$.
However, how do we show that:
1) $W_\alpha$ the subspace depends only on $\alpha$ and not the representation above as an elementary tensor? I'm not sure how to do this one. Should I do so by contradiction? ie assume $\alpha$ has two elementary tensor representations with only one of them generating $W_\alpha$ and then show this is a contradiction, making this a problem about uniqueness of elementary tensor representation?
2) $W_\alpha = W_\beta \iff \exists c \neq 0$ such that $\alpha = c\beta$. The $\implies$ direction is where I'm having issues i.e., the direction where we assume $W_\alpha = W_\beta$. I was considering a proof by contradiction where I induct on $r$.