Vector Scaled by Zero is Zero Vector

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Theorem

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.


Then:

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


Proof

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

$\blacksquare$


Also see


Sources