Chapter 4.5, Problem 86E

Guided Proof Let $S$ be a spanning set for a finite dimensional vector space $V$. Prove that there exists a subset $S^{\prime }$ of $S$ that forms a basis for $V$.

Getting Started: $S$ is a spanning set, but it may not be a basis because it may be linearly dependent. You need to remove extra vectors so that a subset $S^{\prime }$ is a spanning set and is also linearly independent.

(i) If $S$ is linearly independent set, then you are done. If not, remove some vector $v$ from $S$ that is a linear combination of the other vectors in $S$. Call this set $S_1$.

(ii) If $S_1$ is a linearly independent set, then you are done. If not, then continue to remove dependent vectors until you produce a linearly independent subset $S^{\prime }$.

(iii) Conclude that this subset is the minimal spanning set $S^{\prime }$.