Synopses & Reviews
A student-oriented approach to linear algebra, now in its Second Edition
Linear Algebra: Ideas and Applications, Second Edition is an introductory-level linear algebra text for students who require a clear understanding of key algebraic concepts and their applications in such fields as science, engineering, and computer science. The text utilizes a parallel structure that introduces abstract concepts such as linear transformations, eigenvalues, vector spaces, and orthogonality in tandem with computational skills, thereby demonstrating clear and immediate relations between theory and application.
The First Edition was well received, effectively meeting LACSG (Linear Algebra Curriculum Study Group) recommendations in its student-orientation, client-based applications, matrix orientation, and use of technology. The Second Edition, while maintaining the strengths of the original text, has been updated to include input from instructors, students, and other users, and to reflect recent developments in the field.
Important features of the Second Edition include:
- Gradual development of vector spaces
- Highly readable proofs
- Conceptual exercises
- Applications sections for self-study
- Early orthogonality option
- Numerous computer projects using MATLAB and Maple
Synopsis
A student-oriented approach to linear algebra, now in its Second EditionThis introductory-level linear algebra text is for students who require a clear understanding of key algebraic concepts and their applications in such fields as science, engineering, and computer science. The text utilizes a parallel structure that introduces abstract concepts such as linear transformations, eigenvalues, vector spaces, and orthogonality in tandem with computational skills, thereby demonstrating clear and immediate relations between theory and application.
Important features of the Second Edition include:
- Gradual development of vector spaces
- Highly readable proofs
- Conceptual exercises
- Applications sections for self-study
- Early orthogonality option
- Numerous computer projects using MATLAB and Maple
Synopsis
Written for students within a linear algebra class, this text covers a number of applications of linear algebra. It features a unique treatment of vector spaces, proofs and computations, an emphasis on geometry and a selection of computer exercises.
Synopsis
In the author's words: "I wrote this book because I have a deep conviction that mathematics is about ideas, not just formulas and algorithms, and not just theorems and proofs. The text covers the material usually found in a one-semester linear algebra class. It is written. however, from the point of view that knowing why is just as important as knowing how. To ensure that the readers see not only why a given fact is true, but also why it is important, I have included a number of the beautiful applications of linear algebra. Most of my students seem to like this emphasis. For many, mathematics has always been a body of facts to be blindly accepted and used. The notion that they personally can decide mathematical truth or falsehood comes as a revelation. Promoting this level of understanding is the goal of this text." -Richard Penney, from the Preface
About the Author
RICHARD C. PENNEY, PhD, is Professor in the Department of Mathematics and Director of the Mathematics/Statistics Actuarial Science Program at Purdue University, Lafayette, Indiana. He has authored several journal articles and has received several major teaching awards.
Table of Contents
'Preface.
Features of the Text.
1. Systems of Linear Equations.
1.1 The Vector Space of mxnMatrices
1.1.1 Computer Projects.
1.1.2 Applications to Graph Theory 1.
1.2 Systems.
1.2.1 Computer Projects.
1.2.2 Applications to Circuit Theory.
1.3 Gaussian Elimination.
1.3.1 Computer Projects.
1.3.2 Applications to Traffic Flow.
1.4 Column Space and Nullspace.
1.4 1 Computer Projects.
1.4.2 Applications to Predator-Prey Problems.
2. Linear Independence and Dimension.
2.1 The Test for Linear Independence.
2.1.1 Computer Projects.
2.2 Dimension.
2.2.1 Computer Projects.
2.2.2 Applications to Calculus.
2.2.3 Applications to Differential Equations.
2.2.4 Applications to the Harmonic Oscillator.
2.2.5 Computer Projects.
2.3 Row Space and the Rank-Nullity Theorem.
2.3.1 Computer Projects.
3. Linear Transformations.
3.1 The Linearity Properties.
3.1.1 Computer Projects.
3.1.2 Applications to Control Theory.
3.2 Matrix Multiplication (Composition).
3.2.1 Computer Projects.
3.2.2 Applications to Graph Theory II.
3.3 Inverses.
3.3.1 Computer Projects.
3.3.2 Applications to Economics.
3.4 The LU Factorization.
3.4.1 Computer Projects.
3.5 The Matrix of Linear Transformation.
3.5.1 Computer Projects.
4. Determinants.
4.1 Definition of the Determinants.
4.1.1 The Rest of the Proofs.
4.1.2 Computer Projects.
4.2 Reduction and Determinants.
4.2.1 Application to Volume.
4.3 A Formula for Inverses.
5. Eigenvectors and Eigenvalues.
5.1 Eigenvectors.
5.1.1 Computer Projects.
5.1.2 Application to Markov Processes.
5.2 Diagonalization.
5.2.1 Computer Projects.
5.2.2 Applications to Systems of Differential Equations.
5.3 Complex Eigenvectors.
5.3.1 Computer Projects.
6. Orthogonality.
6.1 The Scalar Product in R^{n}.
6.1.1 Application to Statistics.
6.2 Projections: The Gram-Schmidt Process.
6.2.1 Computer Projects.
6.3 Fourier Series: Scalar Product Spaces.
6.3.1 Computer Projects.
6.4 Orthogonal Matrices.
6.5 Least Squares.
6.5.1 Computer Projects.
6.6 Quadratic Forms: Orthogonal Diagonalization.
6.6.1 Computer Projects.
6.7 The Singular Value Decomposition.
Appendix: Answers and Hints.
Glossary.
Index.
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