Basis Theorem

• categories: Linear algebra, Theorem

Statement

Every vector space $V$ over a field $F$ has a Basis

Proof Outline

1. Finite-dimensional case: The proof can proceed by extending any linearly independent set to a spanning set, ensuring that a basis exists for finite-dimensional spaces.
2. Infinite-dimensional case: The existence of a basis for any vector space (even infinite-dimensional) is guaranteed using Zorn’s Lemma, a result equivalent to the Axiom of choice. Zorn’s Lemma ensures that every linearly independent subset of $V$ can be extended to a maximal linearly independent set, which is then a basis for $V$.

Consequences

• For a finite-dimensional vector space $V$ of dimension $n$, every basis of $V$ has exactly $n$ elements.
• For infinite-dimensional spaces, bases can be uncountably infinite, and different bases can have different cardinalities in spaces without finite dimension.