Zero Vector Space Product iff Factor is Zero/Proof 1

Proof
A vector space is a module, so all results about modules also apply to vector spaces.

So from Scalar Product with Identity it follows directly that:
 * $\lambda = 0_F \lor \mathbf v = e \implies \lambda \circ \mathbf v = \bszero$

Next, suppose $\lambda \circ \mathbf v = \bszero$ but $\lambda \ne 0_F$.

Then: