Applied Linear Algebra

For in-depth Linear Algebra courses that focus on applications. This text aims to teach basic methods and algorithms used in modern, real problems that are likely to be encountered by engineering and science students-and to foster understanding of why mathematical techniques work and how they can be derived from first principles. No text goes as far (and wide) in applications. The authors present applications hand in hand with theory, leading students through the reasoning that leads to the important results, and provide theorems and proofs where needed. Because no previous exposure to linear algebra is assumed, the text can be used for a motivated entry-level class as well as advanced undergraduate and beginning graduate engineering/applied math students.

Table of Contents:
Chapter 1. Linear Algebraic Systems 1.1. Solution of Linear Systems 1.2. Matrices and Vectors 1.3. Gaussian Elimination - Regular Case 1.4. Pivoting and Permutations 1.5. Matrix Inverses 1.6. Transposes and Symmetric Matrices 1.7. Practical Linear Algebra 1.8. General Linear Systems 1.9. Determinants Chapter 2. Vector Spaces and Bases 2.1. Vector Spaces 2.2. Subspaces 2.3. Span and Linear Independence 2.4. Bases 2.5. The Fundamental Matrix Subspaces 2.6. Graphs and Incidence Matrices Chapter 3. Inner Products and Norms 3.1. Inner Products 3.2. Inequalities 3.3. Norms 3.4. Positive Definite Matrices 3.5. Completing the Square 3.6. Complex Vector Spaces Chapter 4. Minimization and Least Squares Approximation 4.1. Minimization Problems 4.2. Minimization of Quadratic Functions 4.3. Least Squares and the Closest Point 4.4. Data Fitting and Interpolation Chapter 5. Orthogonality 5.1. Orthogonal Bases 5.2. The Gram-Schmidt Process 5.3. Orthogonal Matrices 5.4. Orthogonal Polynomials 5.5. Orthogonal Projections and Least Squares 5.6. Orthogonal Subspaces Chapter 6. Equilibrium 6.1. Springs and Masses 6.2. Electrical Networks 6.3. Structures Chapter 7. Linearity 7.1. Linear Functions 7.2. Linear Transformations 7.3. Affine Transformations and Isometries 7.4. Linear Systems 7.5. Adjoints Chapter 8. Eigenvalues 8.1. Simple Dynamical Systems 8.2. Eigenvalues and Eigenvectors 8.3. Eigenvector Bases and Diagonalization 8.4. Eigenvalues of Symmetric Matrices 8.5. Singular Values 8.6. Incomplete Matrices and the Jordan Canonical Form Chapter 9. Linear Dynamical Systems 9.1. Basic Solution Methods 9.2. Stability of Linear Systems 9.3. Two-Dimensional Systems 9.4. Matrix Exponentials 9.5. Dynamics of Structures 9.6. Forcing and Resonance Chapter 10. Iteration of Linear Systems 10.1. Linear Iterative Systems 10.2. Stability 10.3. Matrix Norms 10.4. Markov Processes 10.5. Iterative Solution of Linear Systems 10.6. Numerical Computation of Eigenvalues Chapter 11. Boundary Value Problems in One Dimension 11.1. Elastic Bars 11.2. Generalized Functions and the Green's Function 11.3. Adjoints and Minimum Principles 11.4. Beams and Splines 11.5. Sturm-Liouville Boundary Value Problems 11.6. Finite Elements

Review :
Some Quotes from Reviewers "The material on the concept of a general vector space, linear independence, basis, etc. is always difficult for students in this course. This book handles it very well. It gives full, clear explanations. The style is very good, clear, and thorough. It should appeal to my students. I like the book very much. It subscribes to the same philosophy of linear algebra as pioneered by Strang some 30 years ago (acknowledged in the introduction) and builds on the Strang books, making things even clearer and adding more topics. I would certainly like to use this book and would recommend it to my colleagues." -Bruno Harris, Brown University "I like the book very much. We will consider it for our linear algebra courses. This is the best new book to appear since the text by Gilbert Strang. It is really modern book, combining, in a masterful, core and applied aspects of linear algebra. This is a very good book written by a very good mathematician and a very good teacher." -Juan J. Manfredi, University of Pittsburgh "In many, if not most, beginning texts of linear algebra, the applications may be collected together in a chapter at the end of the book or in an appendix, leaving any inclusion of this material to the discretion of the instructor. However, Applied Linear Algebra by Olver and Shakiban completely reverses this procedure with a total integration of the application with the abstract theory. The effect on the reader is quite amazing. The reader slowly begins to realize two main points: (1) how applications generally drive the abstract theory, and (2) how the abstract theory can illuminate the applications, and resolve solutions in very striking ways. This text is easily the best beginning linear algebra text dealing with the applications in an integrated way that I have seen. There is no doubt that this text will be the standard to which all beginning linear algebra texts will be compared. Simply put, this is an absolutely wonderful text!" -Norman Johnson, University of Iowa "I lover the style of this book, especially the fact that you could feel the authors' enthusiasm about the nice mathematics involved in the theory. The examples were very clear and interesting, and they always tried to approach the same problems over and over again as soon sas they had more weapons at their disposal to attack them. I thought this was great, this text introduces the notion of an abstract space very early (still, after Gaussian Elimination) and in a very natural way, then emphasizes along the way over and over again that tremendously. I would absolutely consider this text. I was really taken by the applications and the organization of the materials. I also loved the abundance of exercises and problems." -Tamas Wiandt, Rochester Institute of Technology "This text is very well-written, has lots of examples, and is easy to read and learn from. I'd use it in my Matrix Methods class. There is a good mixture of routine and more advanced examples." -James Curry, University of Colorado-Boulder "I believe the writing style would appeal to my students because of the clarity and the examples, as well as the tone. I am going to consider its use, once I see its final form." -Fabio Augusto Miner, Purdue University