Theorem

Every linearly independent list of vectors in a finite-dimensional vector space can be extended to a basis of the vector space.

Proof

Suppose ${latex.inlineu_{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 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.