Applied Linear Algebra and Matrix Analysis

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Springer Science & Business Media, Aug 14, 2007 - Mathematics - 384 pages
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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
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