An alternate proof of Theorem 1.5 uses the definition of determinant functions.
(a) Note that the vectors forming S make a linearly dependent set if and only if |S| = 0, and check that the result holds in this case.
(b) For the |S| ≠ 0 case, to show that |TS|/|S| = |T| for all transformations, consider the function d: Mn×n → R given by T → |TS|/|S|. Show that d has the first property of a determinant.
(c) Show that d has the remaining three properties of a determinant function.
(d) Conclude that |TS| = |T| ∙ |S|.