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LINEAR ALGEBRA

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MULTILINEAR ALGEBRA

LINEAR ALGEBRA

The linear algebra material below was written for upper division undergraduates with diverse backgrounds who need to learn linear algebra. It is all Creative Commons CC0 1.0. The first three manuscripts contain the Parts I, II and III as separate pdf files.. The rest of the material is broken down into chapters with an overall index and table of contents.

CoverP1Gill CoverP2Gill CoverP3Gill

PART I (274 pages)

PART II (217 pages)

PART III (303 pages)

Creative Commons CC0 1.0

Creative Commons CC0 1.0

Creative Commons CC0 1.0

Comprehensive Introduction to Linear Algebra

Joel G. Broida and S. Gill Williamson

Creative Commons CC0 1.0

pdf ---- Title Pages, Prefaces and Table of Contents (i - xiv)

PART I -- BASIC LINEAR ALGEBRA

pdf -- Part I: Chapter 0 - Foundations (1- 29)

pdf --Part I: Chapter 1 - An Introduction to Groups (30-67)

pdf --Part I: Chapter 2 - Vector Spaces (68-114)

pdf --Part I: Chapter 3 - Linear Equations and Matrices (115-169)

pdf --Part I: Chapter. 4 - Determinants (170-214)

pdf --Part I: Chapter 5 - Linear Transformations and Matrices (215-251)

PART II: POLYNOMIALS AND CANONICAL FORMS

pdf -Part II: Chapter 6 - Polynomials (252-295)

pdf -Part II: Chapter 7- Linear Transformations and Polynomials (296-381)

pdf -Part II: Chapter 8 - Canonical Forms (382-445)

PART III -- OPERATORS AND TENSORS

pdf -Part III: Chapter 9 - Linear Forms (446-489)

pdf -Part III: Chapter 10 - Linear Operators (490-542)

pdf -Part III: Chapter 11 - Multilinear Mappings and Tensors (543-618)

pdf -Part III: Chapter 12 - Hilbert Spaces (619-679)

pdf ---- Appendices and Bibliography (680-726)

pdf - INDEX (727-734)

MULTILINEAR ALGEBRA

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From 1965 to 1972 my research focused on applications of multilinear algebra to combinatorics. After that, I worked mostly on algorithmic combinatorics. The interplay between combinatorics and multilinear algebra is fascinating. Below we attempt to look back at that subject, review some relevant background material, and address some unresolved issues.

The basic lecture notes for this topic were provided by Professor Marvin Marcus from his seminar at UCSB 1965-70. Dr. Tony Trojanowski provided the corrigenda.

(1) Tensor Spaces* (3) Tensor Algebras
(2) Tensor Transformations
(2) Corrigenda (pdf) (3) Corrigenda (pdf)
(4) Grassmann Algebras (5) Clifford Algebras
(6) Representation Theory

These lecture notes were extended and later published as Finite Dimensional Multilinear Algebra, Part I (1973) and Part II (1975) (both out of print):

14 and 18

* Depending on emphasis, we replace (1) Tensor Spaces, with the following:

Tensor spaces - the basics

For the benefit of first year graduate students, I wrote a review of linear algebra which emphasized canonical forms as a training ground for notational skills and proof techniques. This review has been redone in LaTeX with new material added. I wish to thank Dr. Tony Trojanowski for a careful reading and corrections.

Matrix Canonical Forms

notational skills and proof techniques

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During the 1963-64 academic year, Professor Carl Loewner, Stanford University, visited the Department of Mathematics, UCSB. Professor Loewner asked if Marcus's recent work on inequalities for permanents could be used to derive certain inequalites studied by I. Schur in 1918. We present here the results of that discussion.

Recursive Projections of Symmetric Tensors and Marcus's Proof of the Schur Inequality.

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The following article discusses a generalization of the Laplace expansion theorem that possibly relates to future topics in multilinear algebra.

The common-submatrix Laplace expansion

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