Matrix Theory and LINEAR ALGEBRA

Matrix Theory and

LINEAR ALGEBRA

An open text by Peter Selinger

Based on the original text by Lyryx Learning and Ken Kuttler

Creative Commons License (CC BY)

Matrix Theory and Linear Algebra

An open text by Peter Selinger

Based on the original text by Lyryx Learning and Ken Kuttler First edition

CONTRIBUTIONS

Ken Kuttler, Brigham Young University

Ilijas Farah, York University Marie-Andr?e B. Langlois, Dalhousie University

Peter Selinger, Dalhousie University

Bruce Bauslaugh Peter Chow Nathan Friess

Stephanie Keyowski Claude Laflamme Martha Laflamme

Lyryx Learning Team

Jennifer MacKenzie Tamsyn Murnaghan

Bogdan Sava Larissa Stone

Ryan Yee Ehsun Zahedi

Cover photography by Pok Rie

LICENSE

Creative Commons License (CC BY): This text, including the art and illustrations, are available under the Creative Commons license (CC BY), allowing anyone to reuse, revise, remix and redistribute the text.

To view a copy of this license, visit ???? ?? ? ? ?

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Revision history Current revision: Dal 2018 A

This printing: version 7ea4e31 of January 6, 2024

Extensive edits, additions, and revisions have been completed by Peter Selinger and other contributors. Prior extensive edits, additions, and revisions were made by the editorial staff at Lyryx Learning. All new content (text and images) is released under the same license as noted above.

Dal 2018 A Dal 2017 B 2017 A 2016 B 2016 A 2015 A 2012 A

? P. Selinger: updated title and front matter. ? P. Selinger: Extensive revisions and additions. Rewrote most of Chapters 9?11 from scratch.

Added application on perspective rendering, error correcting codes, least squares approximations and curve fitting, an audio demo for Fourier series, sketching of quadratic forms, and principal component analysis.

? P. Selinger, M.B. Langlois: Re-ordered chapters. Extensive revisions to Chapters 1?8. Added new sections on fields, cryptography, geometric interpretation of linear transformations, recurrences, systems of linear differential equations.

? Lyryx: Front matter has been updated including cover, copyright, and revision pages. ? I. Farah: contributed edits and revisions, particularly the proofs in the Properties of Determinants

II: Some Important Proofs section.

? Lyryx: The text has been updated with the addition of subsections on Resistor Networks and the Matrix Exponential based on original material by K. Kuttler.

? Lyryx: New example on Random Walks developed.

? Lyryx: The layout and appearance of the text has been updated, including the title page and newly designed back cover.

? Lyryx: The content was modified and adapted with the addition of new material and several images throughout.

? Lyryx: Additional examples and proofs were added to existing material throughout.

? Original text by K. Kuttler of Brigham Young University. That version is used under Creative Commons license CC BY ( ???? ?? ? ? ?

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?? ? ?????) made possible by funding from The Saylor Foundation's Open Textbook Challenge. See Elementary Linear Algebra for more information and the original version.

Contents

Preface

1

1 Systems of linear equations

3

1.1 Geometric view of systems of equations . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Algebraic view of systems of equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Elementary operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.4 Gaussian elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.5 Gauss-Jordan elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

1.6 Homogeneous systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

1.7 Uniqueness of the reduced echelon form . . . . . . . . . . . . . . . . . . . . . . . . . . 38

1.8 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

1.9 Application: Balancing chemical reactions . . . . . . . . . . . . . . . . . . . . . . . . . 51

1.10 Application: Dimensionless variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

1.11 Application: Resistor networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

2 Vectors in Rn

61

2.1 Points and vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

2.2 Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

2.3 Scalar multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

2.4 Linear combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

2.5 Length of a vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

2.6 The dot product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

2.6.1 Definition and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

2.6.2 The Cauchy-Schwarz and triangle inequalities . . . . . . . . . . . . . . . . . . . . 81

2.6.3 The geometric significance of the dot product . . . . . . . . . . . . . . . . . . . . 82

2.6.4 Orthogonal vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

2.6.5 Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

2.7 The cross product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

2.7.1 Right-handed systems of vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

2.7.2 Geometric description of the cross product . . . . . . . . . . . . . . . . . . . . . 91

2.7.3 Algebraic definition of the cross product . . . . . . . . . . . . . . . . . . . . . . . 92

2.7.4 The box product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

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