Prove that the basis of a subspace of a linear space can be extended to the basis of the whole space

"Engineering Matrix Theory" Zhang Mingchun, Southeast University Press, 27 pages, Theorem 1.2.3

Theorem 1.2.3 Let W be a subspace of n-dimensional linear space V, if α1,...,αr are the basis of W, then there must be (nr) vectors α(r+1),..., αn lets α1,...,αn be the basis of V, that is, the basis of extending W can be the basis of V.

Proof Use mathematical induction for (nr).

When the dimension of V is r, according to the theorem "let V be"