Linear and Geometric Algebra (Geometric Algebra & Calculus)

✍ Scribed by Alan Macdonald

• Publisher: CreateSpace Independent Publishing Platform
• Year: 2011
• Tongue: English
• Leaves: 208
• Category: Library

✦ Synopsis

This textbook for the first undergraduate linear algebra course presents a unified treatment of linear algebra and geometric algebra, while covering most of the usual linear algebra topics.

This is the June 2021 printing, corrected and slightly revised.

Geometric algebra is an extension of linear algebra. It enhances the treatment of many linear algebra topics. And geometric algebra does much more.
Geometric algebra and its extension to geometric calculus unify, simplify, and generalize vast areas of mathematics that involve geometric ideas. They provide a unified mathematical language for many areas of physics, computer science, and other fields.

The book can be used for self study by those comfortable with the theorem/proof style of a mathematics text.

Visit the book’s web site for more information: http://faculty.luther.edu/~macdonal/laga

I commend Alan Macdonald for his excellent book! His exposition is clean and spare. He has done a fine job of engineering a gradual transition from standard views of linear algebra to the perspective of geometric algebra. The book is sufficiently conventional to be adopted as a textbook by an adventurous teacher without getting flack from colleagues. Yet it leads to gems of geometric algebra that are likely to delight thoughtful students and surprise even the most experienced instructors.
-- David Hestenes, Distinguished Research Professor, Arizona State University

✦ Table of Contents

Contents
Preface
To the Student
I Linear Algebra
1 Vectors
1.1 Oriented Lengths
1.2 Rn
2 Vector Spaces
2.1 Vector Spaces
2.2 Subspaces
2.3 Linear Combinations
2.4 Linear Independence
2.5 Bases
2.6 Dimension
3 Matrices
3.1 Matrices
3.2 Systems of Linear Equations
4 Inner Product Spaces
4.1 Oriented Lengths
4.2 Rn
4.3 Inner Product Spaces
4.4 Orthogonality
II Geometric Algebra
5 G3
5.1 Oriented Areas
5.2 Oriented Volumes
5.3 G3
5.4 Generalized Complex Numbers
5.5 Rotations in R3
6 Gn
6.1 Gn
6.2 How Geometric Algebra Works
6.3 Inner and Outer Products
6.4 The Dual
6.5 Product Properties
6.6 Blades as Outer Products
7 Project, Rotate, Reflect
7.1 Project
7.2 Rotate
7.3 Reflect
III Linear Transformations
8 Linear Transformations
8.1 Linear Transformations
8.2 The Adjoint Transformation
8.3 Outermorphisms
8.4 The Determinant
9 Representations
9.1 Matrix Representations
9.2 Eigenvectors
9.3 Invariant Subspaces
9.4 Symmetric Transformations
9.5 Orthogonal Transformations
9.6 Skew Transformations
9.7 Singular Value Decomposition
IV Appendices
A Prerequisites
A.1 Sets
A.2 Logic
A.3 Theorems
A.4 Functions
B Software
B.1 Linear Algebra
B.2 Geometric Algebra
Index