Every linearly independent list of vectors in a finite-dimensional vector space can be extended to a basis of the vector space.
Suppose \({latex.inline[u_{1}, ..., u_{m}](u_{1}, ..., u_{m})} is linearly independent in a finite-dimensional vector space V. Let \){latex.inlinew{1}, ..., w{m}} be a spanning list. Then the list ${latex.inlineu{1}, ..., u{m}, w{1}, ..., w{m}} also spans V. Well, by 1753318207 - Axler 2.30 Every spanning list contains a basis|2.30, this can be reduced to a basis that contains all the u’s. The u’s do not get deleted because they are linearly independent, and the procedure in 2.30 does not delete linearly independent vectors.