Theorem: Let V be a vector space and $S = (v_1, . . . , v_n)$ a spanning sequence of V . Prove that a minimal spanning subsequence of S is linearly independent and a basis of V.
This problem confuses me a bit. What is a minimal spanning subsequence of S?. I tried to google but minimum spanning tree comes out.
Assume to the contrary that a minimal spanning subsequence X of S is linearly dependent, so $\exists v_j \in X$ such that $v_j \in span(X-v_j)$. This implies that there exists a smaller spanning sub-sequence embedded in X which contradicts our assumption of X being a minimal spanning sub sequence. Therefore, X is linearly independent.
If we append a $v_j \in S-X $ in X, then X must become linearly dependent, otherwise if linearly independent then X is not a spanning sub-sequence of S. Since X spans S and S spans V, then X spans V, so X is a basis of V.