Applied linear algebra ana matrix analysis / Thomas S. Shores.

Includes bibliographical references and index.

Linear Systems of Equationsp. 1 Some Examplesp. 1 Notation and a Review of Numbersp. 9 Gaussian Elimination: Basic Ideasp. 21 Gaussian Elimination: General Procedurep. 33 Computational Notes and Projectsp. 46 Matrix Algebrap. 55 Matrix Addition and Scalar Multiplicationp. 55 Matrix Multiplicationp. 62 Applications of Matrix Arithmeticp. 71 Special Matrices and Transposesp. 86 Matrix Inversesp. 101 Basic Properties of Determinantsp. 114 Computational Notes and Projectsp. 129 Vector Spacesp. 145 Definitions and Basic Conceptsp. 145 Subspacesp. 161 Linear Combinationsp. 170 Subspaces Associated with Matrices and Operatorsp. 183 Bases and Dimensionp. 191 Linear Systems Revisitedp. 198 Computational Notes and Projectsp. 208 Geometrical Aspects of Standard Spacesp. 211 Standard Norm and Inner Productp. 211 Applications of Norms and Inner Productsp. 221 Orthogonal and Unitary Matricesp. 233 Change of Basis and Linear Operatorsp. 242 Computational Notes and Projectsp. 247 The Eigenvalue Problemp. 251 Definitions and Basic Propertiesp. 251 Similarity and Diagonalizationp. 263 Applications to Discrete Dynamical Systemsp. 272 Orthogonal Diagonalizationp. 282 Schur Form and Applicationsp. 287 The Singular Value Decompositionp. 291 Computational Notes and Projectsp. 294 Geometrical Aspects of Abstract Spacesp. 305 Normed Spacesp. 305 Inner Product Spacesp. 312 Gram-Schmidt Algorithmp. 323 Linear Systems Revisitedp. 333 Operator Normsp. 342 Computational Notes and Projectsp. 348 Table of Symbolsp. 355 Solutions to Selected Exercisesp. 357 Referencesp. 375 Indexp. 377

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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