This textbook invites students to discover abstract ideas in linear algebra within the context of applications. Diffusion welding and radiography, the two central applications, are introduced early on and used throughout to frame the practical uses of important linear algebra concepts. Students will learn these methods through explorations, which involve making conjectures and answering open-ended questions. By approaching the subject in this way, new avenues for learning the material emerge: For example, vector spaces are introduced early as the appropriate setting for the applied problems covered; and an alternative, determinant-free method for computing eigenvalues is also illustrated. In addition to the two main applications, the authors also describe possible pathways to other applications, which fall into three main areas: Data and image analysis (including machine learning); dynamical modeling; and optimization and optimal design. Several appendices are included as well, one of which offers an insightful walkthrough of proof techniques. Instructors will also find an outline for how to use the book in a course. Additional resources can be accessed on the authors’ website, including code, data sets, and other helpful material. Application-Inspired Linear Algebra will motivate and immerse undergraduate students taking a first course in linear algebra, and will provide instructors with an indispensable, application-first approach.
Author(s): Heather A. Moon, Thomas J. Asaki, Marie A. Snipes
Series: Springer Undergraduate Texts in Mathematics and Technology
Edition: 1
Publisher: Springer
Year: 2022
Language: English
Pages: 549
Tags: Linear Algebra; Linear Algebra Applied; Diffusion Welding; Radiography
Preface
Outline of Text
Using This Text
Exercises
Computational Tools
Ancillary Materials
Acknowledgements
Contents
About the Authors
Introduction To Applications
1.1 A Sample of Linear Algebra in Our World
1.1.1 Modeling Dynamical Processes
1.1.2 Signals and Data Analysis
1.1.3 Optimal Design and Decision-Making
1.2 Applications We Use to Build Linear Algebra Tools
1.2.1 CAT Scans
1.2.2 Diffusion Welding
1.2.3 Image Warping
1.3 Advice to Students
1.4 The Language of Linear Algebra
1.5 Rules of the Game
1.6 Software Tools
1.7 Exercises
Vector Spaces
2.1 Exploration: Digital Images
2.1.1 Exercises
2.2 Systems of Equations
2.2.1 Systems of Equations
2.2.2 Techniques for Solving Systems of Linear Equations
2.2.3 Elementary Matrix
2.2.4 The Geometry of Systems of Equations
2.2.5 Exercises
2.3 Vector Spaces
2.3.1 Images and Image Arithmetic
2.3.2 Vectors and Vector Spaces
2.3.3 The Geometry of the Vector Space mathbbR3
2.3.4 Properties of Vector Spaces
2.3.5 Exercises
2.4 Vector Space Examples
2.4.1 Diffusion Welding and Heat States
2.4.2 Function Spaces
2.4.3 Matrix Spaces
2.4.4 Solution Spaces
2.4.5 Other Vector Spaces
2.4.6 Is My Set a Vector Space?
2.4.7 Exercises
2.5 Subspaces
2.5.1 Subsets and Subspaces
2.5.2 Examples of Subspaces
2.5.3 Subspaces of mathbbRn
2.5.4 Building New Subspaces
2.5.5 Exercises
Vector Space Arithmetic and Representations
3.1 Linear Combinations
3.1.1 Linear Combinations
3.1.2 Matrix Products
3.1.3 The Matrix Equation Ax=b
3.1.4 The Matrix Equation Ax=0
3.1.5 The Principle of Superposition
3.1.6 Exercises
3.2 Span
3.2.1 The Span of a Set of Vectors
3.2.2 To Span a Set of Vectors
3.2.3 Span X is a Vector Space
3.2.4 Exercises
3.3 Linear Dependence and Independence
3.3.1 Linear Dependence and Independence
3.3.2 Determining Linear (In)dependence
3.3.3 Summary of Linear Dependence
3.3.4 Exercises
3.4 Basis and Dimension
3.4.1 Efficient Heat State Descriptions
3.4.2 Basis
3.4.3 Constructing a Basis
3.4.4 Dimension
3.4.5 Properties of Bases
3.4.6 Exercises
3.5 Coordinate Spaces
3.5.1 Cataloging Heat States
3.5.2 Coordinates in mathbbRn
3.5.3 Example Coordinates of Abstract Vectors
3.5.4 Brain Scan Images and Coordinates
3.5.5 Exercises
Linear Transformations
4.1 Explorations: Computing Radiographs and the Radiographic Transformation
4.1.1 Radiography on Slices
4.1.2 Radiographic Scenarios and Notation
4.1.3 A First Example
4.1.4 Radiographic Setup Example
4.1.5 Exercises
4.2 Transformations
4.2.1 Transformations are Functions
4.2.2 Linear Transformations
4.2.3 Properties of Linear Transformations
4.2.4 Exercises
4.3 Explorations: Heat Diffusion
4.3.1 Heat States as Vectors
4.3.2 Heat Evolution Equation
4.3.3 Exercises
4.3.4 Extending the Exploration: Application to Image Warping
4.4 Matrix Representations of Linear Transformations
4.4.1 Matrix Transformations between Euclidean Spaces
4.4.2 Matrix Transformations
4.4.3 Change of Basis Matrix
4.4.4 Exercises
4.5 The Determinants of a Matrix
4.5.1 Determinant Calculations and Algebraic Properties
4.6 Explorations: Re-Evaluating Our Tomographic Goal
4.6.1 Seeking Tomographic Transformations
4.6.2 Exercises
4.7 Properties of Linear Transformations
4.7.1 One-To-One Transformations
4.7.2 Properties of One-To-One Linear Transformations
4.7.3 Onto Linear Transformations
4.7.4 Properties of Onto Linear Transformations
4.7.5 Summary of Properties
4.7.6 Bijections and Isomorphisms
4.7.7 Properties of Isomorphic Vector Spaces
4.7.8 Building and Recognizing Isomorphisms
4.7.9 Inverse Transformations
4.7.10 Left Inverse Transformations
4.7.11 Exercises
Invertibility
5.1 Transformation Spaces
5.1.1 The Nullspace
5.1.2 Domain and Range Spaces
5.1.3 One-to-One and Onto Revisited
5.1.4 The Rank-Nullity Theorem
5.1.5 Exercises
5.2 Matrix Spaces and the Invertible Matrix Theorem
5.2.1 Matrix Spaces
5.2.2 The Invertible Matrix Theorem
5.2.3 Exercises
5.3 Exploration: Reconstruction Without an Inverse
5.3.1 Transpose of a Matrix
5.3.2 Invertible Transformation
5.3.3 Application to a Small Example
5.3.4 Application to Brain Reconstruction
Diagonalization
6.1 Exploration: Heat State Evolution
6.2 Eigenspaces and Diagonalizable Transformations
6.2.1 Eigenvectors and Eigenvalues
6.2.2 Computing Eigenvalues and Finding Eigenvectors
6.2.3 Using Determinants to Find Eigenvalues
6.2.4 Eigenbases
6.2.5 Diagonalizable Transformations
6.2.6 Exercises
6.3 Explorations: Long-Term Behavior and Diffusion Welding Process Termination Criterion
6.3.1 Long-Term Behavior in Dynamical Systems
6.3.2 Using MATLAB/OCTAVE to Calculate Eigenvalues and Eigenvectors
6.3.3 Termination Criterion
6.3.4 Reconstruct Heat State at Removal
6.4 Markov Processes and Long-Term Behavior
6.4.1 Matrix Convergence
6.4.2 Long-Term Behavior
6.4.3 Markov Processes
6.4.4 Exercises
Inner Product Spaces and Pseudo-Invertibility
7.1 Inner Products, Norms, and Coordinates
7.1.1 Inner Product
7.1.2 Vector Norm
7.1.3 Properties of Inner Product Spaces
7.1.4 Orthogonality
7.1.5 Inner Product and Coordinates
7.1.6 Exercises
7.2 Projections
7.2.1 Coordinate Projection
7.2.2 Orthogonal Projection
7.2.3 Gram-Schmidt Process
7.2.4 Exercises
7.3 Orthogonal Transformations
7.3.1 Orthogonal Matrices
7.3.2 Orthogonal Diagonalization
7.3.3 Completing the Invertible Matrix Theorem
7.3.4 Symmetric Diffusion Transformation
7.3.5 Exercises
7.4 Exploration: Pseudo-Inverting the Non-invertible
7.4.1 Maximal Isomorphism Theorem
7.4.2 Exploring the Nature of the Data Compression Transformation
7.4.3 Additional Exercises
7.5 Singular Value Decomposition
7.5.1 The Singular Value Decomposition
7.5.2 Computing the Pseudo-Inverse
7.5.3 Exercises
7.6 Explorations: Pseudo-Inverse Tomographic Reconstruction
7.6.1 The First Pseudo-Inverse Brain Reconstructions
7.6.2 Understanding the Effects of Noise.
7.6.3 A Better Pseudo-Inverse Reconstruction
7.6.4 Using Object-Prior Information
7.6.5 Additional Exercises
Conclusions
8.1 Radiography and Tomography Example
8.2 Diffusion
8.3 Your Next Mathematical Steps
8.3.1 Modeling Dynamical Processes
8.3.2 Signals and Data Analysis
8.3.3 Optimal Design and Decision Making
8.4 How to move forward
8.5 Final Words
A Transmission Radiography and Tomography: A Simplified Overview
A.1 What is Radiography?
A.2 The Incident X-ray Beam
A.3 X-Ray Beam Attenuation
A.4 Radiographic Energy Detection
A.5 The Radiographic Transformation Operator
A.6 Multiple Views and Axial Tomography
A.7 Model Summary
A.8 Model Assumptions
A.9 Additional Resources
B The Diffusion Equation
C Proof Techniques
C.1 Logic
C.2 Proof structure
C.3 Direct Proof
C.4 Contrapositive
C.5 Proof by Contradiction
C.6 Disproofs and Counterexamples
C.7 The Principle of Mathematical Induction
C.8 Etiquette
D Fields
D.1 Exercises
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Index
