Vector Product is Zero only if 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.

Then:


 * $\forall \lambda \in F: \forall \mathbf v \in \mathbf V: \lambda \circ \mathbf v = \bszero \implies \paren {\lambda = 0_F \lor \mathbf v = \mathbf 0}$

where $\bszero \in \mathbf V$ is the zero vector.

Proof
that:


 * $\exists \lambda \in F: \exists \mathbf v \in \mathbf V: \lambda \circ \mathbf v = \bszero \land \lambda \ne 0_F \land \mathbf v \ne \bszero$

which is the negation of the exposition of the theorem.

Utilizing the vector space axioms:

which contradicts the assumption that $\mathbf v \ne \mathbf 0$.

Also see

 * Zero Vector Scaled is Zero Vector
 * Vector Scaled by Zero is Zero Vector
 * Zero Vector Space Product iff Factor is Zero