A first course in linear algebra by k kuttler

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A First Course in Linear Algebra an Open Text

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CONTRIBUTIONS

Ilijas Farah, York University

Ken Kuttler, Brigham Young University

Lyryx Learning Team

Bruce Bauslaugh

Peter Chow

Nathan Friess

Stephanie Keyowski

Claude Laflamme

Martha Laflamme

Jennifer MacKenzie

Tamsyn Murnaghan

Bogdan Sava

Larissa Stone

Ryan Yee

Ehsun Zahedi

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a d v a n c i n g l e a r n i n g

A First Course in Linear Algebra an Open Text

Base Text Revision History

Current Revision: Version 2017 — Revision A

Extensive edits, additions, and revisions have been completed by the editorial staff at Lyryx Learning.

All new content (text and images) is released under the same license as noted above.

2017 A

• 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

2016 B

• 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 7.35 on Random Walks developed.

2016 A • Lyryx: The layout and appearance of the text has been updated, including the title page and newly

designed back cover.

2015 A

• Lyryx: The content was modified and adapted with the addition of new material and several im-

agesthroughout.

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

2012 A

• Original text by K. Kuttler of Brigham Young University. That version is used under Creative Com-

mons license CC BY (https://creativecommons.org/licenses/by/3.0/) made possible by

funding from The Saylor Foundation’s Open Textbook Challenge. See Elementary Linear Algebra for

more information and the original version.

https://creativecommons.org/licenses/by/3.0/
https://www.saylor.org/site/wp-content/uploads/2012/02/Elementary-Linear-Algebra-1-30-11-Kuttler-OTC.pdf
Contents

Contents iii

Preface 1

1 Systems of Equations 3

1.1 Systems of Equations, Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Systems Of Equations, Algebraic Procedures . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.1 Elementary Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2.2 Gaussian Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.2.3 Uniqueness of the Reduced Row-Echelon Form . . . . . . . . . . . . . . . . . . 25

1.2.4 Rank and Homogeneous Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.2.5 Balancing Chemical Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

1.2.6 Dimensionless Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

1.2.7 An Application to Resistor Networks . . . . . . . . . . . . . . . . . . . . . . . . 38

2 Matrices 53

2.1 Matrix Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.1.1 Addition of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

2.1.2 Scalar Multiplication of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 57

2.1.3 Multiplication of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

2.1.4 The i jth Entry of a Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

2.1.5 Properties of Matrix Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . 67

2.1.6 The Transpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

2.1.7 The Identity and Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

2.1.8 Finding the Inverse of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

2.1.9 Elementary Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

2.1.10 More on Matrix Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

2.2 LU Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

2.2.1 Finding An LU Factorization By Inspection . . . . . . . . . . . . . . . . . . . . . 99

2.2.2 LU Factorization, Multiplier Method . . . . . . . . . . . . . . . . . . . . . . . . 100

2.2.3 Solving Systems using LU Factorization . . . . . . . . . . . . . . . . . . . . . . . 101

2.2.4 Justification for the Multiplier Method . . . . . . . . . . . . . . . . . . . . . . . . 102

iii

iv CONTENTS

3 Determinants 107

3.1 Basic Techniques and Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

3.1.1 Cofactors and 2×2 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . 107 3.1.2 The Determinant of a Triangular Matrix . . . . . . . . . . . . . . . . . . . . . . . 112

3.1.3 Properties of Determinants I: Examples . . . . . . . . . . . . . . . . . . . . . . . 114

3.1.4 Properties of Determinants II: Some Important Proofs . . . . . . . . . . . . . . . 118

3.1.5 Finding Determinants using Row Operations . . . . . . . . . . . . . . . . . . . . 123

3.2 Applications of the Determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

3.2.1 A Formula for the Inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

3.2.2 Cramer’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

3.2.3 Polynomial Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

4 Rn 145

4.1 Vectors in Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

4.2 Algebra in Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

4.2.1 Addition of Vectors in Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

4.2.2 Scalar Multiplication of Vectors in Rn . . . . . . . . . . . . . . . . . . . . . . . . 150

4.3 Geometric Meaning of Vector Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

4.4 Length of a Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

4.5 Geometric Meaning of Scalar Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . 159

4.6 Parametric Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

4.7 The Dot Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

4.7.1 The Dot Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

4.7.2 The Geometric Significance of the Dot Product . . . . . . . . . . . . . . . . . . . 170

4.7.3 Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

4.8 Planes in Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

4.9 The Cross Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

4.9.1 The Box Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

4.10 Spanning, Linear Independence and Basis in Rn . . . . . . . . . . . . . . . . . . . . . . . 192

4.10.1 Spanning Set of Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

4.10.2 Linearly Independent Set of Vectors . . . . . . . . . . . . . . . . . . . . . . . . . 194

4.10.3 A Short Application to Chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . 200

4.10.4 Subspaces and Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

4.10.5 Row Space, Column Space, and Null Space of a Matrix . . . . . . . . . . . . . . . 211

4.11 Orthogonality and the Gram Schmidt Process . . . . . . . . . . . . . . . . . . . . . . . . 232

4.11.1 Orthogonal and Orthonormal Sets . . . . . . . . . . . . . . . . . . . . . . . . . . 233