# Vector Space

**Type:** Concept  
**Domain:** Algebra  
**Codex URL:** /mathematics/algebra/vector-space/  
**Entry status:** Live — v1.0 (2026-05-27)

## Summary
A set of vectors, closed under addition and scalar multiplication over a field, satisfying 8 axioms. Foundation of linear algebra.

## Key examples
ℝⁿ, ℂⁿ, matrices Mₘₓₙ, polynomials P(x), continuous functions C[a,b]

## Basis and dimension
A basis is a linearly independent spanning set; all bases have the same size (dimension).

## Historical development
Grassmann (1844) — "extension theory," largely ignored initially. Peano (1888) — abstract axiomatic definition.

## Sources
### Tier 1
- Axler, S. (2015). *Linear Algebra Done Right*. 3rd ed. Springer.
- Halmos, P.R. (1974). *Finite-Dimensional Vector Spaces*. 2nd ed. Springer.
### Tier 2
- Strang, G. (2016). *Introduction to Linear Algebra*. 5th ed. Wellesley-Cambridge Press.

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*Mathematics Codex entry v1.0 — added 2026-05-27 — thecodex.expert/mathematics/*