Introduction to Linear Algebra by Frank M. Stewart
Author:Frank M. Stewart
Language: eng
Format: epub
Publisher: Dover Publications, Inc.
Published: 2019-03-09T16:00:00+00:00
The main theorem of this section is a famous result relating the value of a determinant to those of its minors.
THEOREM 24.17. If is the minor of obtained by striking out its i-th column and j-th row, then
PROOF. Throughout, j stands for a fixed integer.
The idea of the proof is to translate equation (24.2) into the language of alternating forms. The essential tool is the fact that a determinant can be calculated by evaluating the basic form at the vectors which are the columns of the determinant (Theorem 24.5). Because the are N —1 by N —1 determinants we must use as well N.
As usual let be the natural basis for its dual, ϕ the alternating N-form such that ϕ(x1, x2, . . . , xN) = 1 and
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