Applied Linear Algebra and Matrix Analysis (Google eBook)

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