Vector Scaled by Zero is Zero Vector

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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.


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


\(\ds 0_F \circ \mathbf v\) \(=\) \(\ds \paren {0_F + 0_F} \circ \mathbf v\) Field Axiom $\text A3$: Identity for Addition
\(\ds \) \(=\) \(\ds 0_F \circ \mathbf v + 0_F \circ \mathbf v\) Vector Space Axiom $\text V 5$: Distributivity over Scalar Addition
\(\ds \leadsto \ \ \) \(\ds 0_F \circ \mathbf v + \paren {-0_F \circ \mathbf v}\) \(=\) \(\ds \paren {0_F \circ \mathbf v + 0_F \circ \mathbf v} + \paren {-0_F \circ \mathbf v}\) adding $-0_F \circ \mathbf v$ to both sides
\(\ds \) \(=\) \(\ds 0_F \circ \mathbf v + \paren {0_F \circ \mathbf v + \paren {-0_F \circ \mathbf v} }\) Vector Space Axiom $\text V 2$: Associativity
\(\ds \leadsto \ \ \) \(\ds \bszero\) \(=\) \(\ds 0_F \circ \mathbf v + \bszero\) Vector Space Axiom $\text V 4$: Inverses
\(\ds \) \(=\) \(\ds 0_F \circ \mathbf v\) Vector Space Axiom $\text V 3$: Identity


Also see