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Linear Algebra Gems: Assets for Undergraduate Mathematics

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Brand new. We distribute directly for the publisher. Undergraduate linear algebra is both beautiful and replete with connections to applications and the rest of mathematics. The ... core ideas of the undergraduate course may be understood in a variety of clever ways, and there are many valuable ideas that are, historically, just beyond the conceptual reach or time limitations of beginning courses. Because undergraduate linear algebra is so fundamental to the mathematics curriculum, it is often taught by non-specialists. The purpose of the present volume is to place in the hands of teachers, students, and others who enjoy the subject, a broad collection of short items that can enrich the understanding of linear algebra to a wide audience. Most items are accessible to those with modest preparation, even students in the first course, but many will contain pleasant surprises, even to those long interested in the subject. Some articles have been contributed by experts and long-time teachers of linear algebra. Others were selected through an exhaustive search by the editors from many decades of relevant expository journals. Included also are problems that will enrich and intrigue. The 73 articles are organized into 9 sections, with the 120+ problems grouped into categories in a tenth section. This book was prepared as part of a broad contract with the National Science Foundation to improve undergraduate linear algebra education. Read more Show Less

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Overview

Undergraduate Linear Algebra is both beautiful and replete with real world applications and connections to the rest of mathematics. The purpose of the present volume is to enrich the understanding of linear algebra for a wide audience by placing a broad collection of short items in the hands of teachers, students, and others who enjoy the subject. Because undergraduate linear algebra is so fundamental to the mathematics curriculum, it is often taught by non-specialists and specialists alike. Linear Algebra Gems offers to all teachers clever ways in which core ideas can be presented to their students.

Most articles are accessible to those with modest preparation in linear algebra, including beginning students.  However, many items will also contain pleasant surprises even to those well-versed in the subject.   The editors have combed through the literature to find expository articles and problems to enrich the reader's understanding.  The seventy-three articles selected are organized into nine sections, with over 120 problems grouped into subject categories as a tenth section.

Contributors to the volume include experts in the field and long-time teachers of linear algebra.  The book was prepared as part of a broad contract with the National Science Foundation to improve undergraduate linear algebra education. The editors hope that many readers will find enjoyment from this collection.

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Product Details

  • ISBN-13: 9780883851708
  • Publisher: Mathematical Association of America
  • Publication date: 1/28/2002
  • Series: Notes Series
  • Edition description: New Edition
  • Pages: 342
  • Product dimensions: 8.40 (w) x 10.90 (h) x 0.80 (d)

Table of Contents

Foreword
Part 1- Partitioned Matrix Multiplication
Introduction; the Editors
Modern Views of Matrix Multiplication; the Editors
The Associativity of Matrix Multiplications; the Editors
Relationships Between AB and BA; the Editors
The Characteristic Polynomial of a Partitioned Matrix; D. Steven Mackey
The Cauchy-Binet Formula; Wayne Barrett
Cramer's Rule; the Editors
The Multipliacativity of the Determinant; William Watkins and the Editors

Part 2- Determinants
Introduction; the Editors
The Determinant of a Sum of Matrices; L.M. Weiner
Determinants of Sums; Marvin Marcus
An Application of Determinants; Helen Skala
Cramer's Rule via Selective Annihilation; Dan Kalman
The Multiplication of Determinants; D.C. Lewis
A Short Proof of a Result of Determinants; George Marsaglia
Dodgson's Identity; Wayne Barrett
On the Evaluation of Determinants by Chio's Method; L.E. Fuller and J.D. Logan
Apropos Predetermined Determinants; Antal E. Fekete

Part 3-Eigenanalysis
Introduction- the Editors
Eigenvalues, Eigenvectors and Linear Matrix Equations; Charles R. Johnson
Matrices with "Custom Built" Eigenspaces; W.P. Galvin
A Note on Normal Matrices; C.G. Cullen
Eigenvectors: Fixed Vectors and fixed Directions (Discovery Exercises); J. Stuart
On Cauchy's Inequalities for Hermitian Matrices; Emeric Deutsch and Harry Hochstadt
The Monotonicity Theorem, Cauchy's Interlace Theorem and the Courant-Fisher Theorem; Yasuhiko Ikebe, Toshiyuki Inagaki, and Sadaaki Miyamoto
The Power Method for Finding Eigenvalues on a Microcomputer; Gareth Williams and Donna Williams
Singular values and the Spectral Theorem; K. Hoechsmann
The Characteristic Polynomial of a Singular Matrix; D.E. Varberg
Characteristic Roots or Rank 1 Matrices; Larry Cummings
A Method for Finding the Eigenvectors of a n X n Matrix Corresponding to Eigenvalues of Multiplicity One; M. Carchidi

Part 4- Geometry
Introduction; the Editors
Cram-Schmidt Projections; Charles R. Johnson, D. Olesky, and p. van den Driessche
Pythagoras and the Cauchy-Schwarz Inequality; Ladnor Geissinger
An Application of the Schwarz Inequality; V.G. Sigillito
A Geometric Interpretation of Cramer's Rule; Gregory Conner and Michael Lundquist
Isometries of lp-norm; Chi-Kwong Li and Wasin So
Matrices Which Take a Given Vector into a Given Vector; M. Machover
The Matrix of Rotation; Roger C. Alperin
Standard Least Squares Problem; James Foster
Multigrid Graph Paper; Jean H. Bevis

Part 5- Matrix Forms
Introduction; the Editors
LU Factorization; Charles R. Johnson
Singular Value Decomposition: The 2 X 2 Case; Michael Lundquist
A Simple Proof of the Jordan Decomposition Theorem for Matrices; Israel Goldberg and Seymour Goldberg
Similarity of Matrices; William Watkins
Classifying Row-reduced Echelon Matrices; Stewart Venit and Wayne Bishop

Part 6- Polynomials and Matrices
Introduction; the Editors
On the Cayley-Hamilton Theorem; Robert Reams
On Polynomial Matrix Equations; Harley Flanders
The Matrix Equations X squared=A; W.R. Utz
Where Did the Variables Go?; Stephen Barnett
A Zero-Row Reduction Algorithm for Obtaining the gcd of Polynomials; Sidney H. JKung and Yap S. Chua

Part 7- Linear Systems, Inverses and Rank
Introduction; the Editors
Left and Right Inverses; the Editors
Row and column Ranks Are Always Equal; Dave Stanford
A Proof of the Equality of Column and Row Rank of a Matrix; Hans Liebeck
The Frobenius Rank Inequality; Donald Robinson
A New Algorithm for Computing the Rank of a Matrix; Larry Gerstein
Elementary Row Operations and LU Decomposition; David P. Kraines, Vivian Kraines, and David A. Smith
A Succinct Notation for Linear Combinations of Abstract Vectors; Leon Katz
Why Should We Pivot in Gaussian Elimination?; Edward Rozema
The Gaussian Algorithm for Linear Systems with Interval data; G. Alefeld and G. Mayer
On Sylvester's Law of Nullity; Kurt Bing
Inverses of Vandermonde Matrices; F.D. Parker
On One-sided Inverses of Matrices; Elmar Zemgalis
Integer Matrices Whose Inverses Contain Only Integers; Robert Hanson

Part 8- Applications
Introduction; the Editors
The Matrix-Tree Theorem; Ralph P. Grimaldi and Robert J. Lopez
Algebraic Integers and Tensor Products of Matrices; Shaun M. Fallat
On Regular Markov Chains; Nicholas J. Rose
Integration by Matrix Inversion; William Swartz
Some Explicit Formulas for the Exponential Matrix; T.M. Apostol
Avoiding the Jordan Canonical Form in the Discussion of Linear Systems with Constant Coefficients; E.J. Putzer
The Discrete Analogue of the Putzer Algorithm; Saber Elaydi
The Minimum Length of a Permutation as a Product of Transpositions; George Mackiw

Part 9- Other Topics
Introduction; the Editors
Positive Definite Matrices; C.R. Johnson
Quaternions; Robert S. Smith
Bionomial Matrices; Jay Strum
A Combinatorial Theorem on Circulant Matrices; Dean Clark
An Elementary Approach to the Functional Calculus for Matrices; Yasuhiko Ikebe and Toshiyuki Inagaki

Part 10- Problems
Introduction; the Editors
Partitioned Matrix Multiplication
Determinants
Eigenanalysis
Geometry
Matrix Forms
Matrix Equations
Linear Systems, Inverses, and Rank
Applications
Hermitian Matrices
Magic Squares
Special Matrices
Stochastic Matrices
Trace
Other Topics

Author Index
Further Items of Interest

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Foreword


Foreword
This volume, compiled by the editors on behalf of the Linear Algebra Curriculum Study Group, is for instructors and students of linear algebra as well as all those interested in the ideas of elementary linear algebra. We have noticed, through attendance at special sessions organized at the Joint Annual Meetings and through talks given at other conferences and universities, that there is broad and sustained interest in the content of undergraduate linear algebra courses.
    Since the course became a centerpiece of the mathematics curriculum, beginning around 1960, new topics and new treatments have gradually reshaped it, with noticeably greater evolution than in calculus courses. In addition, current courses are often taught by those not trained in the subject or by those who learned linear algebra in a course rather different from the present one. In this setting, it is not surprising that there is considerable interest in the context and subtleties of ideas in the linear algebra course and in a perspective based upon what lies just beyond. With this in mind, we have selected 74 items and an array of problems, some previously published and some submitted in response to our request for such items. We  hope that these will provide a useful background and alternative techniques for instructors, sources of enrichment projects and extended problems for teachers and students, impetus for further textbook evolution to writers, and the enjoyment of discovery to others.
    The Linear Algebra Curriculum Study Group (LACSG) began with a special session, at the January 1990 Joint Annual Meetings, focusing upon the elementary linear algebra course. This session was organized by Duane Porter, following upon an NSF-sponsored Rocky Mountain Mathematics Consortium Lecture Series given by Charles Johnson at the University of Wyoming; David Carlson and David Lay were panel members for that session. With NSF encouragement and support, these four organized a five-day workshop held at the College of William and Mary in August, 1990. The goal was to initiate substantial and sustained national interest in improving the undergraduate linear algebra curriculum. The workshop panel was broadly based, both geographically and with regard to the nature of institutions represented. In addition, consultants from client disciplines described the role of linear algebra in their areas and suggested ways in which the curriculum could be improved from their perspective.
    Preliminary versions of LACSG recommendations were completed at this workshop and widely circulated for common. After receiving comments and with the benefit of much discussion, a version was published in 1993.This was followed by a companion volume to this one in 1997 (Resources for Teaching Linear Algebra). Work of the LACSG has continued with the organization of multiple special sessions at each of the Joint Annual meetings from 1990 through 1998. With sustained strong attendance at those sessions, acknowledged influence on newer textbooks, discussion in letters to the AMS Notices, and the ATLAST workshops, the general goal of the LACSG is being met.
    Though a few items in this volume are several pages, we have generally chosen short, crisp items that we feel contain an interesting idea. Previously published items have been screened from several hundred we reviewed from the past 50+ years, most from the American Mathematical Monthly and the college Mathematics Journal. New (previously unpublished) items were selected from about 100 responses to our call for contributions. Generally, we have chosen items that relate in some way to the first course, might evolve into the first course, or are just beyond it. However, second courses are an important recommendation of the LACSG, and some items are, perhaps, only appropriate at this level. Typically, we have avoided for which both the topic and treatment are well established in current textbooks. For example, there has been dramatic improvement in the last few decades in the use of row operations and reduced echelon form to elucidate or make calculations related to basic issues in linear algebra. But, many of these have quickly found their way into textbooks and become well established, so that we have not included discussion of them. Also, because of the ATLAST  volume, we have not concentrated upon items that emphasize the use of technology in transmitting elementary ideas, though this is quite important. We do not claim that each item is a "gem" in every respect, but something intrigued us about each one. Thus, we feel that each has something to offer and, also, that every reader will find something of interest.
    Based upon what we found, the column is organized into ten topical "parts." The parts and the items within each part are in no particular order, except that we have tried to juxtapose items that are closely related. Many items do relate to parts other than the one we chose. The introduction to each part provides a bit of background or emphasizes important issues about constituent items. Because of the number of items reprinted without editing we have not adopted a common notation. Each item should be regarded as a stand-alone piece with its own notation or utilizing fairly standard notation.
 Each item is attributed at the end of the item, typically on one of three ways. If it is a reprinted item, the author, affiliation at the time, and original journal reference are given. If it is an original contribution, the author and affiliation are given. In a few cases, the editors chose to author a discussion they felt important, and such items are simply attributed to "the Editors."
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