{"id":"vector-space","domain":"algebra","type":"Concept","title":"Vector Space","slug":"vector-space","url":"/mathematics/algebra/vector-space/","summary":"A set of vectors closed under addition and scalar multiplication (over a field), satisfying 8 axioms. Foundation of linear algebra.","formal_definition":"V over field F with vector addition and scalar multiplication satisfying associativity, commutativity, identity, inverses, and distributive laws.","key_concepts":["Basis and dimension","Linear transformation","Subspace","Inner product space","Hilbert space"],"added_to_codex":"2026-05-27","last_verified":"2026-05-27","sources":[{"tier":1,"citation":"Axler, S. (2015). Linear Algebra Done Right. 3rd ed. Springer."},{"tier":1,"citation":"Halmos, P.R. (1974). Finite-Dimensional Vector Spaces. 2nd ed. Springer."},{"tier":2,"citation":"Strang, G. (2016). Introduction to Linear Algebra. 5th ed. Wellesley-Cambridge Press."}],"relationships":[{"type":"defined_over","target":"field","label":"Field"},{"type":"generalises","target":"group","label":"Group"},{"type":"uses","target":"function","label":"Function"},{"type":"contains","target":"polynomial","label":"Polynomial"}],"entry_status":"live","reading_levels_available":["curious","exploring","deep_dive"],"completeness":1.0,"codex_version":"1.4"}