**Courant Lecture Notes**

Volume: 29;
2019;
261 pp;
Softcover

MSC: Primary 97; 15; 40; 20;

**Print ISBN: 978-1-4704-4871-4
Product Code: CLN/29**

List Price: $51.00

AMS Member Price: $40.80

MAA Member Price: $45.90

**Electronic ISBN: 978-1-4704-5149-3
Product Code: CLN/29.E**

List Price: $51.00

AMS Member Price: $40.80

MAA Member Price: $45.90

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# Linear Algebra I

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*Frederick P. Greenleaf; Sophie Marques*

A co-publication of the AMS and the Courant Institute of Mathematical Sciences at New York University

This book is the first of two volumes on
linear algebra for graduate students in mathematics, the sciences, and
economics, who have: a prior undergraduate course in the subject; a
basic understanding of matrix algebra; and some proficiency with
mathematical proofs. Proofs are emphasized and the overall objective
is to understand the structure of linear operators as the key to
solving problems in which they arise.

This first volume re-examines basic notions of linear algebra:
vector spaces, linear operators, duality, determinants,
diagonalization, and inner product spaces, giving an overview of
linear algebra with sufficient mathematical precision for advanced use
of the subject. This book provides a nice and varied selection of
exercises; examples are well-crafted and provide a clear understanding
of the methods involved. New notions are well motivated and
interdisciplinary connections are often provided, to give a more
intuitive and complete vision of linear algebra. Computational aspects
are fully covered, but the study of linear operators remains the focus
of study in this book.

Titles in this series are co-published with the Courant Institute of Mathematical Sciences at New York University.

#### Readership

Undergraduate and graduate students interested in an advanced understanding of linear algebra for graduate studies.

#### Reviews & Endorsements

The book is nicely written at an appropriate level of sophistication, and there is a good selection of exercises, both imbedded in the text itself and collected, section by section, at the end of each chapter.

-- Mark Hunacek, Iowa State University

Students would benefit immensely from a course based on a textbook like this one.

-- Fernando Gouvêa, MAA Reviews

#### Table of Contents

# Table of Contents

## Linear Algebra I

- Cover Cover11
- Title page iii4
- Contents v6
- Preface ix10
- Chapter 1. Vector Spaces over a Field π 112
- 1.1. Vector Spaces 112
- 1.2. Vector Subspaces 718
- 1.3. Solving Matrix Equation π΄π₯=π 1223
- Elementary Operations and Echelon Form of π΄π₯=π 1324
- The Homogeneous Equation π΄π₯=0 1627
- Solving Inhomogeneous Equations π΄π₯=π 1728
- Determining the Linear Span of a Set of Vectors 1829
- Implicit and Parametric Descriptions of Vector Subspaces 1930
- More on Elementary Row and Column Operations 2031

- 1.4. Linear Span, Independence, and Bases 2132
- 1.5. Quotient Spaces π/π 3142
- Additional Exercises 3849
- Appendix: The Degree Formula for π [π₯β,β¦,π₯_π] 4657

- Chapter 2. Linear Operators π:πβπ 4960
- Chapter 3. Duality and the Dual Space π* 89100
- 3.1. Definitions and Examples 89100
- 3.2. Dual Bases in the Dual Space π* 94105
- 3.3. The Transpose π^π : π*βπ* of π : πβπ 97108
- Matrix Description of a Transpose π^π 98109
- Calculating the Transpose of a Projection Operator 99110
- Reflexivity of Finite-Dimensional Spaces 101112
- The Annihilator πΒ° of a Subspace π in π 103114
- A Dimension Formula for Annihilators πΒ° 103114
- Row Rank vs. Column Rank (Revisited) 104115
- Outline of a Proof That (RowRank) = (ColRank) 104115

- Additional Exercises 105116

- Chapter 4. Determinants 109120
- 4.1. The Permutation Group π_π 109120
- 4.2. Determinants 118129
- Proving the Basic Properties of πππ‘(π΄) 119130
- Row Operations, Determinants, and Matrix Inverses 122133
- Computing Matrix Inverses 124135
- Computational Issues 126137
- Proving the Multiplicative Property πππ‘(π΄π΅)=πππ‘(π΄)πππ‘(π΅) 127138
- Defining Determinants of Linear Operators 129140
- More on Rank, RowRank, and ColumnRank 130141
- Expansion by Minors and Cramerβs Rule 132143

- Additional Exercises 134145

- Chapter 5. The Diagonalization Problem 139150
- 5.1. Eigenvalues, Characteristic Polynomial, and Spectrum 139150
- 5.2. Eigenvalues and the Characteristic Polynomial 145156
- 5.3. Diagonalization and Limits of Operators 153164
- Norms on Finite-Dimensional Spaces 153164
- Multiplicative Properties of Norms on Matrix Space 154165
- The Operator Norm ||π||_{ππ} on Linear Operators and Matrices 155166
- Equivalence of Norms on Finite-Dimensional Spaces 157168
- Practical Calculations with Matrix Norms 158169
- Convergence of Sequences and Series in Matrix Space 158169
- Properties of Matrix Norms 159170

- 5.4. Application: Computing the Exponential π^{π΄} of a Matrix 160171
- 5.5. Application: Linear Systems of Differential Equations 165176
- 5.6. Application: Matrix-Valued Geometric Series 169180
- Additional Exercises 172183

- Chapter 6. Inner Product Spaces 181192
- 6.1. Basic Definitions and Examples 181192
- 6.2. Orthogonal Complements and Projections 190201
- 6.3. Adjoints and Orthonormal Decompositions 200211
- Diagonalization vs. Orthogonal Diagonalization 200211
- Dual Spaces of Inner Product Spaces 201212
- The Adjoint Operator π*:πβπ 202213
- Linear Projections vs. Orthogonal Projections 204215
- Adjoint π* vs. Transpose π^{π} 206217
- Computing an Operator Adjoint 206217
- Self-Adjoint, Unitary, and Normal Operators. 207218

- 6.4. Diagonalization in Inner Product Spaces 209220
- Orthogonal Diagonalization 209220
- Schur Normal Form 209220
- Diagonalizing Self-Adjoint and Normal Operators 212223
- Unitary Equivalence of Operators vs. Similarity 216227
- The Matrix Groups π(π), ππ(π), π(π), ππ(π) 219230
- Change of Orthonormal Basis 221232
- Diagonalization over π = β: A Summary 222233

- 6.5. Reflections, Rotations, and Rigid Motions on ββΏ 222233
- 6.6. Spectral Theorem for Vector and Inner Product Spaces 229240
- 6.7. Positive Operators and Polar Decomposition 239250
- Additional Exercises 248259

- Index 257268
- Back Cover Back Cover1275