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Linear Algebra, Geometry and Transformation


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Table of Contents

Vectors, Mappings and Linearity
Numeric Vectors
Mappings and Transformations
The Matrix of a Linear Transformation

Solving Linear Systems
The Linear System
The Augmented Matrix and RRE Form
Homogeneous Systems in RRE Form
Inhomogeneous Systems in RRE Form
The Gauss-Jordan Algorithm
Two Mapping Answers Linear Geometry
Geometric Vectors
Geometric/Numeric Duality
Dot-Product Geometry
Lines, Planes, and Hyperplanes
System Geometry and Row/Column Duality The Algebra of Matrices
Matrix Operations
Special Matrices
Matrix Inversion
A Logical Digression
The Logic of the Inversion Algorithm
Determinants Subspaces
Basic Examples and Definitions
Spans and Perps
Perp/Span Conversion
Dimension and Rank Orthogonality
Four Subspaces, 16 Questions
Orthonormal Bases
The Gram-Schmidt Algorithm Linear Transformation
Kernel and Image
The Linear Rank Theorem
Eigenvalues and Eigenspaces: Calculation
Eigenvalues and Eigenspaces: Similarity
Diagonalizability and the Spectral Theorem
Singular Value DecompositionAppendix A: Determinants
Appendix B: Proof of the Spectral Theorem
Appendix C: Lexicon Index

About the Author

Bruce Solomon is a professor in the Department of Mathematics at Indiana University Bloomington, where he often teaches linear algebra. He has held visiting positions at Stanford University and in Australia, France, and Israel. His research articles explore differential geometry and geometric variational problems. He earned a PhD from Princeton University.


"All the standard topics of a first course are covered, but the treatment omits abstract vector spaces. ... What is unusual is the author's aim to interpret every concept and result geometrically, thus motivating the student to learn to visualize what is going on, rather than just relying on calculations. This is a strong and useful feature. ... The book has very many practice sections with over 500 exercises, most of them numerical. ... As the author mentions in the preface, it was his aim to provide a sound mathematical introduction, and in the reviewer's opinion he has succeeded in doing this."
-Zentralblatt MATH 1314

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