Vector Inverse is Negative Vector

Theorem

Let $F$ be a field whose zero is $0_F$ and whose unity is $1_F$.

Let $\struct {\mathbf V, +, \circ}_F$ be a vector space over $F$, as defined by the vector space axioms.

Then:

$\forall \mathbf v \in \mathbf V: -\mathbf v = -1_F \circ \mathbf v$

Proof

\(\ds \mathbf v + \paren {-1_F \circ \mathbf v}\)

\(=\)

\(\ds \paren {1_F \circ \mathbf v} + \paren {-1_F \circ \mathbf v}\)

Field Axiom $\text M3$: Identity for Product

\(\ds \)

\(=\)

\(\ds \paren {1_F + \paren {- 1_F} } \circ \mathbf v\)

Vector Space Axiom $\text V 5$: Distributivity over Scalar Addition

\(\ds \)

\(=\)

\(\ds 0_F \circ \mathbf v\)

Field Axiom $\text A4$: Inverses for Addition

\(\ds \)

\(=\)

\(\ds \mathbf 0\)

Vector Scaled by Zero is Zero Vector

so $-1_F \circ \mathbf v$ is an additive inverse of $\mathbf v$.

From Additive Inverse in Vector Space is Unique:

$-1_F \circ \mathbf v = -\mathbf v$

$\blacksquare$

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 $64 \ \text{(v)}$