# Zero Vector Scaled is Zero Vector

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## 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.

## Proof

 $\ds \lambda \circ \bszero$ $=$ $\ds \lambda \circ \paren {\bszero + \bszero}$ Vector Space Axiom $\text V 3$: Identity $\ds$ $=$ $\ds \lambda \circ \bszero + \lambda \circ \bszero$ Vector Space Axiom $\text V 6$: Distributivity over Vector Addition $\ds \leadsto \ \$ $\ds \lambda \circ \bszero + \paren {-\lambda \circ \bszero}$ $=$ $\ds \paren {\lambda \circ \bszero + \lambda \circ \bszero} + \paren {-\lambda \circ \bszero}$ adding $-\lambda \circ \bszero$ to both sides $\ds$ $=$ $\ds \lambda \circ \bszero + \paren {\lambda \circ \bszero + \paren {-\lambda \circ \bszero} }$ Vector Space Axiom $\text V 2$: Associativity $\ds \leadsto \ \$ $\ds \bszero$ $=$ $\ds \lambda \circ \bszero + \bszero$ Vector Space Axiom $\text V 4$: Inverses $\ds$ $=$ $\ds \lambda \circ \bszero$ Vector Space Axiom $\text V 3$: Identity

$\blacksquare$