Let $V$ and $W$ be vector spaces over $\mathbb{C}$. Show that $0\otimes w = v\otimes 0 = 0 \in V \otimes W$. Algebraically, the vector space $V \otimes W$ is spanned by elements of the form $v \otimes w$, and the following rules are satisfied, for any scalar $c$. The definition is the same no matter which scalar field is used.
(1) $c(v_1 \otimes w_1) = (cv_1) \otimes w_1 = v_1 \otimes (cw_1)$
(2) $(v_1 + v_2) \otimes w_1 = v_1 \otimes w_1 + v_2 \otimes w_1$
(3) $v_1 \otimes (w_1 + w_2) = v_1 \otimes w_1 + v_1 \otimes w_2$
One basic consequence of these formulas is that
$0\otimes w = v\otimes0 = 0$.
How would I go about proving that $0\otimes w = v\otimes0 = 0$?
Thank you.

Hint: Use formula (1) with $c=0$.
