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
| \(\ds 0_F \circ \mathbf v\) | \(=\) | \(\ds \paren {0_F + 0_F} \circ \mathbf v\) | 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 |
$\blacksquare$
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
Sources
- 1965: Claude Berge and A. Ghouila-Houri: Programming, Games and Transportation Networks ... (previous) ... (next): $1$. Preliminary ideas; sets, vector spaces: $1.2$. Vector Spaces
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $7$: Vector Spaces: $\S 32$. Definition of a Vector Space: Theorem $64 \ \text{(iii)}$