Vector Scaled by Zero is Zero Vector

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

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

Then:


 * $\forall \mathbf v \in \mathbf V: 0_F \circ \mathbf v = \bszero$

Also see

 * Zero Vector Scaled is Zero Vector
 * Vector Product is Zero only if Factor is Zero
 * Zero Vector Space Product iff Factor is Zero
 * Vector Inverse is Negative Vector