Let $V$ and $W$ be vector spaces over $\mathbb{C}$. Show that $0\otimes w = v\otimes 0 = 0 \in V \otimes W$.

Question

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.

Answer 1

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