Differential Geometry by Marcelo Epstein

Author:Marcelo Epstein
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

3.4.4.5 Linear Independence, Decomposability, Determinant

Since we will not need the full apparatus of exterior algebra, we content ourselves with the statement of a few essential facts.3

Proposition 3.1.

A set of vectors v 1 ,…, v k in V is linearly dependent if, and only if, their wedge product vanishes.

Proof.

If one of the vectors v i is expressible as a linear combination of the others, each of the wedge products resulting by using the distributive property vanishes, since there would always be a repeated factor. Conversely, if the given vectors are linearly independent, we can complete the set by adding m − r further linearly independent vectors so as to form a base of V. The m-fold wedge product of all the elements of this basis is itself a basis of Λ m (V ) (whose dimension is 1) and, therefore it cannot vanish, nor can any partial product thereof.

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.