Every spanning list in a vector space can be reduced to a basis of the vector.
Suppose ${latex.inlinev{1},...v{n}} spans V. We want to remove some of the vectors from the list so that the remaining vectors form a basis of V. We do this through the multistep process described below:
Step 1 If \({latex.inline[v_{1} = 0](v_{1} = 0)} delete it. If \){latex.inlinev_{1} \neq 0}, leave the list unchanged.
Step k If \({latex.inline[v_{k} \in span(v_{1}, ..., v_{k-1})](v_{k} \in span(v_{1}, ..., v_{k-1}))}, delete it from the list. If \){latex.inlinev_{k}}is not in the span, leave B unchanged.
Stop the process after step n. The process ensures that no ${latex.inlinev_{k}} is in the span of the previous v’s. Thus it is linearly independent per linear dependence lemma. The span also did not change since we only removed vectors that had extra info. Thus, we have a basis.