**Courant Lecture Notes**

Volume: 30;
2020;
288 pp;
Softcover

MSC: Primary 97; 15; 40; 20;

**Print ISBN: 978-1-4704-5425-8
Product Code: CLN/30**

List Price: $59.00

AMS Member Price: $47.20

MAA Member Price: $53.10

**Electronic ISBN: 978-1-4704-5642-9
Product Code: CLN/30.E**

List Price: $59.00

AMS Member Price: $47.20

MAA Member Price: $53.10

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# Linear Algebra II

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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 second 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. Both volumes have been used for several years in
a one-year course sequence, Linear Algebra I and II, offered at New
York University's Courant Institute.

The first three chapters of this second volume round out the
coverage of traditional linear algebra topics: generalized
eigenspaces, further applications of Jordan form, as well as bilinear,
quadratic, and multilinear forms. The final two chapters are
different, being more or less self-contained accounts of special
topics that explore more advanced aspects of modern algebra: tensor
fields, manifolds, and vector calculus in Chapter 4 and matrix Lie
groups in Chapter 5. The reader can choose to pursue either
chapter. Both deal with vast topics in contemporary mathematics. They
include historical commentary on how modern views evolved, as well as
examples from geometry and the physical sciences in which these topics
are important.

The book provides a nice and varied selection of exercises;
examples are well-crafted and provide a clear understanding of the
methods involved.

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

#### Table of Contents

# Table of Contents

## Linear Algebra II

- Cover Cover11
- Contents v6
- Preface ix10
- Chapter 1. Generalized Eigenspaces and the Jordan Decomposition 116
- An Overview of This Chapter 116
- 1.1. Nilpotent Operators and Examples 318
- 1.2. Fine Structure of Nilpotent Operators 924
- 1.3. Generalized Eigenspaces 2136
- 1.4. The Generalized Eigenspace Decomposition 2540
- 1.5. Consequences of the Jordan Canonical Form (JCF) 3752
- Additional Exercises 4661
- 1.6. Appendix A: Brief Review of Diagonalization 5570

- Chapter 2. Further Applications of Jordan Form 6378
- An Overview of This Chapter 6378
- 2.1. The Jordan Form and Differential Equations 6479
- 2.2. Normal Forms for Linear Operators over ℝ 7590
- Complexification: A Case Study 7691
- Complexification of Real Vector Spaces and Operators 7792
- Relations Between RR [𝑡] and CC [𝑡] 7893
- Complexification of Linear Operators over RR 8095
- Subspaces of Real Type in 𝑉_{CC } 8398
- Concordance Between Eigenspaces of symbfit 𝑇 in symbfit 𝑉 and of symbfit {𝑇}_{symbf {CC }} in symbfit {𝑉}_{CC } 85100
- The Normal Form of a Linear Operator Over RR 88103
- When symbfit {𝑇_{CC }} Is Not Diagonalizable 89104

- Additional Exercises 94109

- Chapter 3. Bilinear, Quadratic, and Multilinear Forms 99114
- An Overview of This Chapter 99114
- 3.1. Basic Definitions and Examples 100115
- 3.2. Canonical Models for Bilinear Forms 106121
- The Automorphism Group of a Bilinear Form 𝐵 107122
- Canonical Forms. Case 1: symbfit {𝐵} Symmetric, KK =RR 109124
- Sylvester’s Theorem: Invariance of the Signature on 𝑂𝑠𝑦𝑚𝑏𝑓𝑖𝑡(𝑝,𝑞) 113128
- A Diagonalization Algorithm 115130
- The Gauss-Seidel Algorithm 117132
- Canonical Forms. Case 2: symbfit 𝐵 Symmetric, symbfit {KK =CC } 118133
- Canonical Forms: Case 3: symbfit {𝐵} Antisymmetric, KK =RR or CC 120135

- 3.3. Sesquilinear Forms (KK =CC ) 122137
- Additional Exercises 127142

- Chapter 4. Tensor Fields, Manifolds, and Vector Calculus 131146
- An Overview of This Chapter 131146
- 4.1. Tangent Vectors, Cotangent Vectors, and Tensors 134149
- Smooth Functions and Mappings on Manifold symbfit {𝑀} 138153
- Tangent Vectors and the Tangent Spaces TM _{𝑝} 141156
- Some Objections: Finding the Right Definition 142157
- Change of Coordinates 147162
- Vector Fields as Differential Operators on symbfit 𝑀 149164
- The Differential of a Map Between Manifolds 151166

- 4.2. Cotangent Vectors and Differential Forms 154169
- 4.3. Differential Forms on symbfit 𝑀 and Their Exterior Derivatives 161176
- Action of Permutation Groups symbfit {𝑆_{𝑛}} on Tensors 165180
- Products of Tensors and Tensor Fields 167182
- Wedge Product of Antisymmetric Tensors 167182
- Calculating Wedge Products on a Chart 170185
- Exterior Derivative of Higher Rank sbfi 𝑘-Forms 172187
- Primitives of sbfi 𝑘-Forms. 175190
- Poincaré’s Lemma 176191
- Transferring Calculations Between 𝑀 and RR ^{𝑚} 177192
- Proof of Poincaré’s Lemma 178193

- 4.4. Div, Grad, Curl, and All That 181196
- Additional Exercises 189204

- Chapter 5. Matrix Lie Groups 203218
- An Overview of This Chapter 203218
- 5.1. Matrix Groups and the Implicit Function Theorem 204219
- Smooth Mappings and Their Differentials 205220
- Smooth Hypersurfaces and the Implicit Function Theorem (IFT) 207222
- Rank vs Dimension of Level Sets 211226
- The “Maximal Rank” Case 212227
- The Inverse Mapping Theorem (IMT) 213228
- Differentiable Manifolds (General Definition) 214229
- Differentiable Manifolds and the IFT 215230

- 5.2. Matrix Lie Groups 221236
- 5.3. Lie Algebra Structure in Tangent Spaces of Lie Groups 242257
- 5.4. The Exponential Map for Matrix Lie Groups 254269
- One-Parameter Subgroups in Matrix Lie Groups 255270
- The Logarithm Logb (symbfit {𝐴)} of a Matrix 256271
- Singularities of the Exponential Map on symbf {𝑀(n,CC )} 258273
- Relation Between symbf {𝐸𝑥𝑝: rk {𝑔𝑙}}→symbf {𝐺𝐿} and Its Restriction symbf {𝑒𝑥𝑝: rk {𝑔}→𝐺}. 260275
- Case Study: Nilpotent Lie Groups 261276
- The Maps 𝐸𝑥𝑝,𝐴𝑑, 𝑎𝑑 Revisited 264279
- Connected Lie Groups 270285
- The Campbell-Baker-Hausdorff Formula: Recovering symbfit 𝐺 from rk {𝑔} 272287

- 5.5. The Lie Correspondence: Lie Groups vs Lie Algebras 274289
- Additional Exercises 277292

- Bibliography 289304
- Back Cover Back Cover1307