Solomon Linear Algebra, Geometry and Transformation

1. Auflage 2014
ISBN: 978-1-4822-9930-4
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 474 Seiten

Reihe: Textbooks in Mathematics

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.

Autoren/Hrsg.

Solomon, Bruce

Fachgebiete

• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Angewandte Mathematik, Mathematische Modelle
• Mathematik | Informatik Mathematik Algebra

Inhaltsverzeichnis

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

Über Autor(innen)

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.