Application-inspired linear algebra / / Heather A. Moon, Thomas J. Asaki, Marie A. Snipes

• Record Nr.: UNISA996479369703316
• Autore: Moon Heather A.
• Pubbl/distr/stampa: Cham, Switzerland : , : Springer, , [2022]; ©2022
• ISBN: 3-030-86155-4
• Descrizione fisica: 1 online resource (538 pages)
• Collana: Springer Undergraduate Texts in Mathematics and Technology
• Disciplina: 512.5
• Soggetti: Algebras, Linear; Àlgebra lineal; Llibres electrònics
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Note generali: Includes index.
• Nota di contenuto: Intro -- 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.