Vector Scaled by Zero is Zero Vector

Theorem
Let $\left({\mathbf V, +, \circ}\right)_{\mathbb F}$ be a vector space over $\mathbb F$, as defined by the vector space axioms.

Then:


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

where $0 \in \mathbb F$ is the zero scalar.

Proof
Utilizing the vector space axioms:

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