Linear Algebra, Geometry and Transformation: 1st Edition (e-Book) book cover

Linear Algebra, Geometry and Transformation

1st Edition

By Bruce Solomon

Chapman and Hall/CRC

474 pages | 63 B/W Illus.

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Description

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.

Reviews

"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

Table of Contents

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

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.

About the Series

Textbooks in Mathematics

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Subject Categories

BISAC Subject Codes/Headings:
MAT002000
MATHEMATICS / Algebra / General
MAT003000
MATHEMATICS / Applied