© 2015-2016 Jacob Ström, Kalle Åström, and Tomas Akenine-Möller

Immersive Linear Algebra: Table of Contents



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Preface

1 Introduction
1.1 Brief Note on Notation
1.2 Some Trigonometry

2 Vectors
2.1 Points and Vectors
2.2 Vector Addition
2.3 Scalar Vector Multiplication
2.4 Properties of Vector Arithmetic
2.5 Vector Bases and Coordinates
2.6 Vector Spaces with More Than 3 Dimensions
2.6.1 The General Definition
2.7 Summary

3 The Dot Product
3.1 Introduction
3.2 Definition and Applications
3.2.1 Unit Vectors and Normalization
3.2.2 Projection
3.2.3 Rules and Properties
3.3 Orthonormal Basis
3.3.1 Vector Length in Orthonormal Basis
3.4 Inequalities
3.5 Some Examples
3.6 Lines and Planes
3.6.1 Lines
3.6.2 Planes
3.7 Follow up on Ray Tracing

4 The Vector Product
4.1 Introduction
4.2 Orientation
4.3 Definition of Vector Product
4.4 Rules and Properties
4.5 Scalar Triple Product
4.6 Vector Triple Product
4.7 Examples
4.8 Followup on the Introduction Example

5 Gaussian Elimination
5.1 Introduction
5.2 Examples
5.3 Gaussian Elimination
5.4 Special Cases
5.5 The Homogeneous Case
5.6 Implicit and Explicit form
5.7 Theoretical Underpinning
5.7.1 The Gaussian Elimination Rules
5.7.2 The General Case
5.8 Linear Dependence and Independence
5.9 Spanning
5.10 Change of Basis

6 The Matrix
6.1 Introduction
6.2 Definition
6.3 Matrix Operations
6.3.1 Matrix Multiplication by a Scalar
6.3.2 Matrix Addition
6.3.3 Matrix-Matrix Multiplication
6.4 Some Useful Two- and Three-Dimensional Matrices
6.4.1 Two Dimensions
6.4.2 Three Dimensions
6.5 Properties of Matrix Arithmetic
6.6 Matrix Inverse
6.7 Inverses, Independence, and Span
6.8 Change of Base
6.9 Orthogonal Matrices
6.10 Followup on the Introduction Example

7 Determinants
7.1 Introduction
7.2 Definition
7.3 Permutations and Determinants
7.4 Transposition, Multiplication, and Inverse
7.5 Expansion Along a Column
7.6 Adjoint Matrix
7.7 Cramer's Rule
7.8 Determinants, Independence, and Invertibility

8 Rank
8.1 Linear Subspaces
8.2 Null Space and Nullity
8.3 Column Space, Row Space, and Rank
8.4 Rank and Determinants
8.5 Followup on the Introduction Example

9 Linear Mappings

10 Eigenvalues and Eigenvectors

11 Matrix Factorizations