Vectors, Mappings and Linearity
Numeric Vectors
Functions
Mappings and Transformations
Linearity
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
Nullspace
Column-Space
Perp/Span Conversion
Independence
Basis
Dimension and Rank
Orthogonality
Orthocomplements
Four Subspaces, 16 Questions
Orthonormal Bases
The Gram-Schmidt Algorithm
Linear Transformation
Kernel and Image
The Linear Rank Theorem
Eigenspaces
Eigenvalues and Eigenspaces: Calculation
Eigenvalues and Eigenspaces: Similarity
Diagonalizability and the Spectral Theorem
Singular Value Decomposition
Appendix A: Determinants
Appendix B: Proof of the Spectral Theorem
Appendix C: Lexicon
Index
Biography
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
