Multilinear AlgebraThis book is a revised version of the first edition and is intended as a Linear Algebra sequel and companion volume to the fourth edition of (Graduate Texts in Mathematics 23). As before, the terminology and basic results of Linear Algebra are frequently used without refer~nce. In particular, the reader should be familiar with Chapters 1-5 and the first part of Chapter 6 of that book, although other sections are occasionally used. In this new version of Multilinear Algebra, Chapters 1-5 remain essen tially unchanged from the previous edition. Chapter 6 has been completely rewritten and split into three (Chapters 6, 7, and 8). Some of the proofs have been simplified and a substantial amount of new material has been added. This applies particularly to the study of characteristic coefficients and the Pfaffian. The old Chapter 7 remains as it stood, except that it is now Chapter 9. The old Chapter 8 has been suppressed and the material which it con tained (multilinear functions) has been relocated at the end of Chapters 3, 5, and 9. The last two chapters on Clifford algebras and their representations are completely new. In view of the growing importance of Clifford algebras and the relatively few references available, it was felt that these chapters would be useful to both mathematicians and physicists. |
Contents
Tensor Products | 1 |
Tensor Products of Vector Spaces with Additional Structure | 41 |
Tensor Algebra | 60 |
Copyright | |
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antiderivation arbitrary associative algebra assume b₁ basis bilinear function C₁ C₂ Clifford algebra commutative consider the linear Corollary define a linear definition degree zero denote direct sum dual spaces E₁ E₂ Euclidean space exterior algebra F₁ finite dimension follows formula given graded algebra Hence homogeneous of degree homomorphism implies induced injective inner product space involution L(Rē Lemma Linear Algebra linear isomorphism linear map linear transformation multiplication n-dimensional nondegenerate obtain operator orthogonal p-linear pair of dual PROOF Proposition prove relation representation restriction satisfies scalar product Section skew skew-symmetric subspace symmetric tensor algebra tensor product Theorem u₁ unique unit element universal property v₁ vector space ɅPE w₁ w₂ whence x₁ xp+q y₁ ye F yields ΛΕ