Zero Vector Scaled is Zero Vector

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

Then:


 * $\forall \lambda \in \mathbb F: \lambda \circ \bszero = \bszero$

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

Also see

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