Applied linear algebra and matrix methods
 Responsibility
 Timothy G. Feeman.
 Publication
 Cham : Springer, 2024.
 Physical description
 1 online resource (330 p.).
 Series
 Springer undergraduate texts in mathematics and technology.
Online
More options
Description
Creators/Contributors
 Author/Creator
 Feeman, Timothy G., 1956
Contents/Summary
 Contents

 Intro
 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 ``All1s'' 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 kMeans
 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 InputOutput 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 ColumnRow 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 TwoState 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 kMeans
 9.6 Drawing a Graph
 Summary
 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 everincreasing 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 intext 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. .
Subjects
 Subjects
 Algebras, Linear.
 Algèbre linéaire.
 Genre
 Electronic books.
Bibliographic information
 Publication date
 2024
 Series
 Springer Undergraduate Texts in Mathematics and Technology
 ISBN
 9783031395628 (electronic bk.)
 303139562X (electronic bk.)
 3031395611
 9783031395611
 DOI
 10.1007/9783031395628