Applied Linear Algebra and Matrix Methods

✍ Scribed by Timothy G. Feeman

• Publisher: Springer
• Year: 2023
• Tongue: English
• Leaves: 330
• Series: Springer Undergraduate Texts in Mathematics and Technology
• Edition: 1
• Category: Library

✦ Synopsis

This textbook is designed for a first course in linear algebra for undergraduate students from a wide range of quantitative and data driven fields. By focusing on applications and implementation, students will be prepared to go on to apply the power of linear algebra in their own discipline. With an ever-increasing need to understand and solve real problems, this text aims to provide a growing and diverse group of students with an applied linear algebra toolkit they can use to successfully grapple with the complex world and the challenging problems that lie ahead. Applications such as least squares problems, information retrieval, linear regression, Markov processes, finding connections in networks, and more, are introduced on a small scale as early as possible and then explored in more generality as projects. Additionally, the book draws on the geometry of vectors and matrices as the basis for the mathematics, with the concept of orthogonality taking center stage. Important matrix factorizations as well as the concepts of eigenvalues and eigenvectors emerge organically from the interplay between matrix computations and geometry.

The R files are extra and freely available. They include basic code and templates for many of the in-text examples, most of the projects, and solutions to selected exercises. As much as possible, data sets and matrix entries are included in the files, thus reducing the amount of manual data entry required.

✦ Table of Contents

Introduction
Advice for Instructors
Acknowledgments
Contents
1 Vectors
1.1 Coordinates and Vectors
1.2 The Vector Norm
1.3 Angles and the Inner Product
1.4 Inner Product and Vector Arithmetic
1.5 Statistical Correlation
1.6 Information Retrieval
1.6.1 Comparing Movie Viewers
1.7 Distance on a Sphere
1.8 Bézier Curves
1.9 Orthogonal Vectors
1.10 Area of a Parallelogram
1.11 Projection and Reflection
1.12 The ``All-1s'' Vector
1.13 Exercises
1.14 Projects
2 Matrices
2.1 Matrices
2.1.1 Algebraic Properties of Matrix Arithmetic
2.2 Matrix Multiplication
2.2.1 Algebraic Properties of Matrix Multiplication
2.3 The Identity Matrix, I
2.4 Matrix Inverses
2.5 Transpose of a Matrix
2.6 Exercises
3 Matrix Contexts
3.1 Digital Images
3.2 Information Retrieval Revisited
3.3 Markov Processes: A First Look
3.4 Graphs and Networks
3.5 Simple Linear Regression
3.6 k-Means
3.7 Projection and Reflection Revisited
3.8 Geometry of 22 Matrices
3.9 The Matrix Exponential
3.10 Exercises
3.11 Projects
4 Linear Systems
4.1 Linear Equations
4.2 Systems of Linear Equations
4.3 Row Reduction
4.4 Row Echelon Forms
4.5 Matrix Inverses (And How to Find Them)
4.6 Leontief Input–Output Matrices
4.7 Cubic Splines
4.8 Solutions to AX=B
4.9 LU Decomposition
4.10 Affine Projections
4.10.1 Kaczmarz's Method
4.10.2 Fixed Point of an Affine Transformation
4.11 Exercises
4.12 Projects
5 Least Squares and Matrix Geometry
5.1 The Column Space of a Matrix
5.2 Least Squares: Projection into Col(A)
5.3 Least Squares: Two Applications
5.3.1 Multiple Linear Regression
5.3.2 Curve Fitting with Least Squares
5.4 Four Fundamental Subspaces
5.4.1 Column–Row Factorization
5.5 Geometry of Transformations
5.6 Matrix Norms
5.7 Exercises
5.8 Project
6 Orthogonal Systems
6.1 Projections Revisited
6.2 Building Orthogonal Sets
6.3 QR Factorization
6.4 Least Squares with QR
6.5 Orthogonality and Matrix Norms
6.6 Exercises
6.7 Projects
7 Eigenvalues
7.1 Eigenvalues and Eigenvectors
7.2 Computing Eigenvalues
7.3 Computing Eigenvectors
7.4 Transformation of Eigenvalues
7.5 Eigenvalue Decomposition
7.6 Population Models
7.7 Rotations of R3
7.8 Existence of Eigenvalues
7.9 Exercises
8 Markov Processes
8.1 Stochastic Matrices
8.2 Stationary Distributions
8.3 The Power Method
8.4 Two-State Markov Processes
8.5 Ranking Web Pages
8.6 The Monte Carlo Method
8.7 Random Walks on Graphs
8.8 Exercises
8.9 Project
9 Symmetric Matrices
9.1 The Spectral Theorem
9.2 Norm of a Symmetric Matrix
9.3 Positive Semidefinite Matrices
9.3.1 Matrix Square Roots
9.4 Clusters in a Graph
9.5 Clustering a Graph with k-Means
9.6 Drawing a Graph
9.7 Exercises
9.8 Projects
10 Singular Value Decomposition
10.1 Singular Value Decomposition
10.2 Reduced Rank Approximation
10.3 Image Compression
10.4 Latent Semantic Indexing
10.5 Principal Component Analysis
10.6 Least Squares with SVD
10.7 Approximate Least Squares Solutions
10.8 Exercises
10.9 Projects
Bibliography
Index

✦ Subjects

Linear Algebra; Applied Linear Algebra; Matrix Algebra; Singular Value Decomposition; QR Factorization; Eigenvalues; Eigenvectors