1st Edition

# Linear Algebra, Geometry and Transformation

474 Pages 63 B/W Illustrations
by Chapman & Hall

474 Pages
by Chapman & Hall

Also available as eBook on:

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.

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

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