The Dark Art of Linear Algebra: An Intuitive Geometric Approach

✍ Scribed by Seth Braver

• Publisher: Vector Vectorum Books
• Year: 2023
• Tongue: English
• Leaves: 178
• Category: Library

✦ Table of Contents

Front Matter
Cover
Title Page
Copyright Page
Table of Contents
Acknowledgements
Preface for Teachers
Preface for Students
Chapter 1: Vectors
Adding Vectors
Stretching Vectors
Standard Basis Vectors
Lengths of Vectors
The Dot Product
Chapter 2: Vocabulary
Linear Combinations, Span, Linear Independence
Bases and Subspaces
Parametric Representations of Subspaces (and Affine Spaces)
Chapter 3: Linear Transformations and Matrices
Linear Maps and their Matrices
Non-square Matrices
Another Look at the Matrix-Vector Product
Matrix Multiplication
Matrix Addition and Transposition
Abstract Linear Algebra: A Trailer
Chapter 4: Gaussian Elimination
Linear Systems
Augmented Matrices
Row Operations
Gaussian Elimination (Endgames and Algorithms)
The Matrix Inversion Algorithm
Linear Relations Among the Columns
Image and Kernel
Why 'Kernel' is an Apt Name
Chapter 5: The Determinant
The Determinant: Definition
The Determinant: Properties
Elementary Matrices (and their Determinants)
Computing Determinants (by Row Reduction)
Computing Determinants (by Cofactor Expansion)
Chapter 6: Change of Basis
Coordinates (and Changing Bases)
Matrices of Maps (via Nonstandard Bases)
Linear Isometries, Orthogonal Matrices
Chapter 7: Eigenstuff
Eigenvectors and their Eigenvalues
Finding the Eigenstuff
Eigenstuff and Long Run Behavior
Chapter 8: Projections and Least Squares
Orthogonal Projection
The Gram-Schmidt Process
Best-Fit Lines, Least-Squares Solutions
Transpose to the Rescue
Selected Answers to Exercises
Ch. 1 Answers
Ch. 2 Answers
Ch. 3 Answers
Ch. 4 Answers
Ch. 5 Answers
Ch. 6 Answers
Ch. 7 Answers
Ch. 8 Answers
Index
About the Author