Introduction To Linear Algebra: Computation, Application, and Theory

✍ Scribed by Mark J. DeBonis

• Publisher: CRC Press
• Year: 2022
• Tongue: English
• Leaves: 435
• Series: Textbooks in Mathematics
• Category: Library

✦ Synopsis

Introduction to Linear Algebra: Computation, Application and Theory is designed for students who have never been exposed to the topics in a linear algebra course. The text is filled with interesting and diverse application sections but is also a theoretical text which aims to train students to do succinct computation in a knowledgeable way. After completing the course with this text, the student will not only know the best and shortest way to do linear algebraic computations but will also know why such computations are both effective and successful.

Features

• Includes cutting edge applications in machine learning and data analytics.
• Suitable as a primary text for undergraduates studying linear algebra.
• Requires very little in the way of pre-requisites.

✦ Table of Contents

Cover
Half Title
Series Page
Title Page
Copyright Page
Dedication
Contents
Preface
Chapter 1: Examples of Vector Spaces
1.1. FIRST VECTOR SPACE: TUPLES
1.2. DOT PRODUCT
1.3. APPLICATION: GEOMETRY
1.4. SECOND VECTOR SPACE: MATRICES
1.4.1. Special Matrix Families
1.5. MATRIX MULTIPLICATION
Chapter 2: Matrices and Linear Systems
2.1. SYSTEMS OF LINEAR EQUATIONS
2.2. GAUSSIAN ELIMINATION
2.3. APPLICATION: MARKOV CHAINS
2.4. APPLICATION: THE SIMPLEX METHOD
2.5. ELEMENTARY MATRICES AND MATRIX EQUIVALENCE
2.6. INVERSE OF A MATRIX
2.7. APPLICATION: THE SIMPLEX METHOD REVISITED
2.8. HOMOGENEOUS/NON-HOMOGENEOUS SYSTEMS AND RANK
2.9. DETERMINANT
2.10. APPLICATIONS OF THE DETERMINANT
2.11. APPLICATION: LU FACTORIZATION
Chapter 3: Vector Spaces
3.1. DEFINITION AND EXAMPLES
3.2. SUBSPACE
3.3. LINEAR INDEPENDENCE
3.4. SPAN
3.5. BASIS AND DIMENSION
3.6. SUBSPACES ASSOCIATED WITH A MATRIX
3.7. APPLICATION: DIMENSION THEOREMS
Chapter 4: Linear Transformations
4.1. DEFINITION AND EXAMPLES
4.2. KERNEL AND IMAGE
4.3. MATRIX REPRESENTATION
4.4. INVERSE AND ISOMORPHISM
4.4.1. Background
4.4.2. Inverse
4.4.3. Isomorphism
4.5. SIMILARITY OF MATRICES
4.6. EIGENVALUES AND DIAGONALIZATION
4.7. AXIOMATIC DETERMINANT
4.8. QUOTIENT VECTOR SPACE
4.8.1. Equivalence Relations
4.8.2. Introduction to Quotient Spaces
4.8.3. Applications of Quotient Spaces
4.9. DUAL VECTOR SPACE
Chapter 5: Inner Product Spaces
5.1. DEFINITION, EXAMPLES AND PROPERTIES
5.2. ORTHOGONAL AND ORTHONORMAL
5.3. ORTHOGONAL MATRICES
5.3.1. Definition and Results
5.3.2. Application: Rotations and Reflections
5.4. APPLICATION: QR FACTORIZATION
5.5. SCHUR TRIANGULARIZATION THEOREM
5.6. ORTHOGONAL PROJECTIONS AND BEST APPROXIMATION
5.7. REAL SYMMETRIC MATRICES
5.8. SINGULAR VALUE DECOMPOSITION
5.9. APPLICATION: LEAST SQUARES OPTIMIZATION
5.9.1. Overdetermined Systems
5.9.2. Best Fitting Polynomial
5.9.3. Linear Regression
5.9.4. Underdetermined Systems
5.9.5. Approximating Functions
Chapter 6: Applications in Data Analytics
6.1. INTRODUCTION
6.2. DIRECTION OF MAXIMAL SPREAD
6.3. PRINCIPAL COMPONENT ANALYSIS
6.4. DIMENSIONALITY REDUCTION
6.5. MAHALANOBIS DISTANCE
6.6. DATA SPHERING
6.7. FISHER LINEAR DISCRIMINANT FUNCTION
6.8. LINEAR DISCRIMINANT FUNCTIONS IN FEATURE SPACE
6.9. MINIMAL SQUARE ERROR LINEAR DISCRIMINANT FUNCTION
Chapter 7: Quadratic Forms
7.1. INTRODUCTION TO QUADRATIC FORMS
7.2. PRINCIPAL MINOR CRITERION
7.3. EIGENVALUE CRITERION
7.4. APPLICATION: UNCONSTRAINED NON-LINEAR OPTIMIZATION
7.5. GENERAL QUADRATIC FORMS
Appendix A: Regular Matrices
Appendix B: Rotations and Reflections in Two Dimensions
Appendix C: Answers to Selected Exercises
References
Index