The Essentials of a First Linear Algebra Course and More
Linear Algebra, Geometry and Transformation provides students with a solid geometric grasp of linear transformations. It stresses the linear case of the inverse function and rank theorems and gives a careful geometric treatment of the spectral theorem.
An Engaging Treatment of the Interplay among Algebra, Geometry, and Mappings
The text starts with basic questions about images and pre-images of mappings, injectivity, surjectivity, and distortion. In the process of answering these questions in the linear setting, the book covers all the standard topics for a first course on linear algebra, including linear systems, vector geometry, matrix algebra, subspaces, independence, dimension, orthogonality, eigenvectors, and diagonalization.
A Smooth Transition to the Conceptual Realm of Higher Mathematics
This book guides students on a journey from computational mathematics to conceptual reasoning. It takes them from simple "identity verification" proofs to constructive and contrapositive arguments. It will prepare them for future studies in algebra, multivariable calculus, and the fields that use them.
Print Versions of this book also include access to the ebook version.
Table of Contents
Vectors, Mappings and Linearity
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
Lines, Planes, and Hyperplanes
System Geometry and Row/Column Duality
The Algebra of Matrices
A Logical Digression
The Logic of the Inversion Algorithm
Basic Examples and Definitions
Spans and Perps
Dimension and Rank
Four Subspaces, 16 Questions
The Gram-Schmidt Algorithm
Kernel and Image
The Linear Rank Theorem
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
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