# Applied Linear Algebra and Matrix Analysis

Springer Science & Business Media, Aug 14, 2007 - Mathematics - 384 pages
This book is about matrix and linear algebra, and their applications. 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. To this end, applied mathematics and mathematical modeling ought to have an important role in an introductory treatment of linear algebra. In this way students see that concepts of matrix and linear algebra make concrete problems workable. In this book we weave signi?cant motivating examples into the fabric of the text. I hope that instructors will not omit this material; that would be a missed opportunity for linear algebra! The text has a strong orientation toward numerical computation and applied mathematics, which means that matrix analysis plays a central role. All three of the basic components of l- ear algebra — theory, computation, and applications — receive their due. The proper balance of these components gives students the tools they need as well as the motivation to acquire these tools. Another feature of this text is an emphasis on linear algebra as an experimental science; this emphasis is found in certain examples, computer exercises, and projects. Contemporary mathematical software make ideal “labs” for mathematical experimentation. Nonetheless, this text is independent of speci?c hardware and software pl- forms. Applications and ideas should take center stage, not software.

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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