Preface
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Preface
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1 Introduction
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1.1 Brief Note on Notation
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1.2 Some Trigonometry
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2 Vectors
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2.1 Points and Vectors
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2.2 Vector Addition
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2.3 Scalar Vector Multiplication
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2.4 Properties of Vector Arithmetic
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2.5 Vector Bases and Coordinates
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2.6 Vector Spaces with More Than 3 Dimensions
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2.6.1 The General Definition
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2.7 Summary
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3 The Dot Product
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3.1 Introduction
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3.2 Definition and Applications
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3.2.1 Unit Vectors and Normalization
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3.2.2 Projection
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3.2.3 Rules and Properties
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3.3 Orthonormal Basis
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3.3.1 Vector Length in Orthonormal Basis
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3.4 Inequalities
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3.5 Some Examples
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3.6 Lines and Planes
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3.6.1 Lines
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3.6.2 Planes
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3.7 Follow up on Ray Tracing
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4 The Vector Product
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4.1 Introduction
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4.2 Orientation
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4.3 Definition of Vector Product
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4.4 Rules and Properties
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4.5 Scalar Triple Product
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4.6 Vector Triple Product
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4.7 Examples
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4.8 Followup on the Introduction Example
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5 Gaussian Elimination
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5.1 Introduction
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5.2 Examples
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5.3 Gaussian Elimination
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5.4 Special Cases
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5.5 The Homogeneous Case
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5.6 Implicit and Explicit form
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5.7 Theoretical Underpinning
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5.7.1 The Gaussian Elimination Rules
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5.7.2 The General Case
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5.8 Linear Dependence and Independence
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5.9 Spanning
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5.10 Change of Basis
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6 The Matrix
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6.1 Introduction
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6.2 Definition
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6.3 Matrix Operations
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6.3.1 Matrix Multiplication by a Scalar
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6.3.2 Matrix Addition
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6.3.3 Matrix-Matrix Multiplication
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6.4 Some Useful Two- and Three-Dimensional Matrices
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6.4.1 Two Dimensions
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6.4.2 Three Dimensions
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6.5 Properties of Matrix Arithmetic
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6.6 Matrix Inverse
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6.7 Inverses, Independence, and Span
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6.8 Change of Base
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6.9 Orthogonal Matrices
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6.10 Followup on the Introduction Example
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7 Determinants
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7.1 Introduction
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7.2 Definition
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7.3 Permutations and Determinants
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7.4 Transposition, Multiplication, and Inverse
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7.5 Expansion Along a Column
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7.6 Adjoint Matrix
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7.7 Cramer's Rule
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7.8 Determinants, Independence, and Invertibility
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8 Rank
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8.1 Linear Subspaces
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8.2 Null Space and Nullity
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8.3 Column Space, Row Space, and Rank
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8.4 Rank and Determinants
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8.5 Followup on the Introduction Example
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9 Linear Mappings
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9.1 Introduction
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9.2 Transformation Matrices
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9.3 Composite Linear Mappings
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9.4 Inverse Mapping
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10 Eigenvalues and Eigenvectors
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10.1 Introduction
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10.2 Eigenvalues and Eigenvectors
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10.3 Calculating Eigenvalues and Eigenvectors
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10.4 Diagonalization
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10.5 Diagonalization of symmetric matrices
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10.6 Max Elongation of a Vector During Linear Mapping
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10.7 Miscellaneous Results on Eigenvalues and Eigenvectors
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10.8 Eigenvalues and Eigenvectors are Useful
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10.9 Outlook
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