5.13. Determinants and Independence
A set of n vectors
The negation of Theorem 5.8, which says
thus proving the “if” part of the theorem.
To prove the “only
if” part, let V be the linear n-space in question. Since
it forms a basis for V. By Theorem 4.12, there exists a linear
is the kth
unit coordinate vector. In terms of
components in the Ij
basis, we have
where Ak and Ik are row matrices.
By Lemma 5.10, we have
where A is a matrix with rows
i.e., A is nonsingular and by Theorem