##### Theorem BS Basis of a Span

Suppose that \$S=\set{\vectorlist{v}{n}}\$ is a set of column vectors. Define \$W=\spn{S}\$ and let \$A\$ be the matrix whose columns are the vectors from \$S\$. Let \$B\$ be the reduced row-echelon form of \$A\$, with \$D=\set{\scalarlist{d}{r}}\$ the set of column indices corresponding to the pivot columns of \$B\$. Then

1. \$T=\set{\vect{v}_{d_1},\,\vect{v}_{d_2},\,\vect{v}_{d_3},\,\ldots\,\vect{v}_{d_r}}\$ is a linearly independent set.
2. \$W=\spn{T}\$.

Proof
(in context)