Introduction

The bare essentials of free, tensor, and exterior algebras were given in Chapter 5. This appendix gives two additional applications of finite dimensional exterior algebras. The first one is a necessary and sufficient condition on a finite set of vectors in the underlying vector space V to be independent. The other one is the Laplace expansion of a determinant. From there on we drop the restrictive hypothesis on V that it be finite dimensional, and give a different and far more detailed account of the tensor algebra on V. Then an alternate unified construction of all the basic algebras as quotients of the tensor algebra is given. Thus the tensor algebra serves as a central focus for building various algebras, and in this way this appendix gives a new perspective.

Exterior algebras

Since this appendix is an addendum to and also a continuation of Chapter 5, we simply will continue the numbering of paragraphs from Chapter 5, and assume that the reader is familiar with 5-1.11 through 5-2.20.

The symbol Sr will be used to denote the group of all permutations of {1,2, …, r}; Sr is also frequently called the symmetric group on r elements.