Zero Vector Space Product iff Factor is Zero

Theorem
Let $F$ be a field whose zero is $0_F$ and whose unity is $1_F$.

Let $\struct {\mathbf V, +, \circ}_F$ be a vector space over $F$, as defined by the vector space axioms.

Let $\mathbf v \in \mathbf V, \lambda \in F$.

Then:
 * $\lambda \circ \mathbf v = \bszero \iff \paren {\lambda = 0_F \lor x = \bszero}$

Also see

 * Basic Results about Modules
 * Basic Results about Unitary Modules