📐 Basis Extension Theorem
Every linearly independent set in a finite-dimensional vector space can be extended to a basis.
Proof: Let $S = \\{v_1, \\ldots, v_k\\}$ be linearly independent in $V$ with $\\dim(V) = n$.
If $\\text{span}(S) = V$, then $S$ is already a basis.
Otherwise, there exists $v_{k+1} \\in V \\setminus \\text{span}(S)$. Then $\\{v_1, \\ldots, v_k, v_{k+1}\\}$ is linearly independent.
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From: Advanced Linear Algebra
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