# Tag Archives: Linear Algebra

## Every nontrivial vector space has a basis

Theorem. Let $$V$$ be a vector space and $$M$$ a linearly independent subset of $$V.$$ Then there exists a basis of $$V$$ containing $$M$$, i.e., $$M$$ can be extended to a basis. Proof. The set $$[ M ]$$ of all linearly independent subsets of $$V$$ containing $$M$$ is not empty and is ordered by set-theoric inclusion.… Read More »