# Applied Linear Algebra and Matrix Analysis

Springer Science & Business Media, Aug 14, 2007 - Mathematics - 383 pages

This new book offers a fresh approach to matrix and linear algebra by providing a balanced blend of applications, theory, and computation, while highlighting their interdependence. Intended for a one-semester course, Applied Linear Algebra and Matrix Analysis places special emphasis on linear algebra as an experimental science, with numerous examples, computer exercises, and projects. While the flavor is heavily computational and experimental, the text is independent of specific hardware or software platforms.

Throughout the book, significant motivating examples are woven into the text, and each section ends with a set of exercises. The student will develop a solid foundation in the following topics

*Gaussian elimination and other operations with matrices

*basic properties of matrix and determinant algebra

*standard Euclidean spaces, both real and complex

*geometrical aspects of vectors, such as norm, dot product, and angle

*eigenvalues, eigenvectors, and discrete dynamical systems

*general norm and inner-product concepts for abstract vector spaces

For many students, the tools of matrix and linear algebra will be as fundamental in their professional work as the tools of calculus; thus it is important to ensure that students appreciate the utility and beauty of these subjects as well as the mechanics. By including applied mathematics and mathematical modeling, this new textbook will teach students how concepts of matrix and linear algebra make concrete problems workable.

Thomas S. Shores is Professor of Mathematics at the University of Nebraska, Lincoln, where he has received awards for his teaching. His research touches on group theory, commutative algebra, mathematical modeling, numerical analysis, and inverse theory.

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### Contents

 LINEAR SYSTEMS OF EQUATIONS 1 12 Notation and a Review of Numbers 9 Basic Ideas 21 General Procedure 33 15 Computational Notes and Projects 46 MATRIX ALGEBRA 55 22 Matrix Multiplication 62 23 Applications of Matrix Arithmetic 71
 43 Orthogonal and Unitary Matrices 233 44 Change of Basis and Linear Operators 242 45 Computational Notes and Projects 247 THE EIGENVALUE PROBLEM 251 52 Similarity and Diagonalization 263 53 Applications to Discrete Dynamical Systems 272 54 Orthogonal Diagonalization 282 55 Schur Form and Applications 287

 24 Special Matrices and Transposes 86 25 Matrix Inverses 101 26 Basic Properties of Determinants 114 27 Computational Notes and Projects 129 VECTOR SPACES 145 32 Subspaces 161 33 Linear Combinations 170 34 Subspaces Associated with Matrices and Operators 183 35 Bases and Dimension 191 36 Linear Systems Revisited 198 37 Computational Notes and Projects 208 GEOMETRICAL ASPECTS OF STANDARD SPACES 211 42 Applications of Norms and Inner Products 221
 56 The Singular Value Decomposition 291 57 Computational Notes and Projects 294 GEOMETRICAL ASPECTS OF ABSTRACT SPACES 305 62 Inner Product Spaces 312 63 GramSchmidt Algorithm 323 64 Linear Systems Revisited 333 65 Operator Norms 342 66 Computational Notes and Projects 348 Table of Symbols 355 Solutions to Selected Exercises 357 References 375 Index 377 Copyright