Let $\displaystyle \beta$ be a basis for a vector space $\displaystyle V$ and let $\displaystyle S$ be a linear independent subset of $\displaystyle V$. Prove there exists a subset of $\displaystyle \beta$,$\displaystyle S_1$, such that $\displaystyle S\cup{S_1}$ is a basis for $\displaystyle V$

Proof

Let $\displaystyle S_1$ be the set of all elements in $\displaystyle \beta$ such that $\displaystyle S\union{S_1}$ is linearly independent. Now consider $\displaystyle S\cup{S_1}$

Note that every vector in $\displaystyle S$ can be expressed as a linear combination of vectors in $\displaystyle \beta$. So it follows that every vector in $\displaystyle V$ can be expressed as a linear combination of vectors in $\displaystyle S\cup{S_1}$

Thus $\displaystyle S\cup{S_1}$ is linearly independent and generates $\displaystyle V$, so it is a basis.

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