Zero Vector Scaled 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:


 * $\forall \lambda \in \mathbb F: \lambda \circ \mathbf 0 = \mathbf 0$

where $\mathbf 0 \in \mathbf V$ is the zero vector.

Proof
Utilizing the vector space axioms:

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