Linear Algebra: An Inquiry-Based Approach

✍ Scribed by Jeff Suzuki

• Year: 2021
• Tongue: English
• Leaves: 376
• Category: Library

✦ Table of Contents

Cover
Half Title
Series Page
Title Page
Copyright Page
Dedication
Contents
Introduction and Features
For the Student . . . and Teacher
Prerequisites
Suggested Sequences
1. Tuples and Vectors
1.1. Tuples
Activity 1.1: Equality
1.2. Vectors
Activity 1.2: Feature Vectors
Activity 1.3: Vectors
Activity 1.4: Document Vectors
Activity 1.5: Vector Addition
Activity 1.6: Scalar Multiplication
Activity 1.7: Componentwise Multiplication?
1.3. Proofs
Activity 1.8: Evidence Collection
Activity 1.9: Properties of Vector Arithmetic
Activity 1.10: More Vector Properties
1.4. Directed Distances
Activity 1.11: Directed Distances
Activity 1.12: More Directed Distances
Activity 1.13: Vectors in R5000
Activity 1.14: The Geometry of Vectors
Activity 1.15: Direction and Magnitude
Activity 1.16: Vector Arithmetic
Activity 1.17: Vector Equation of a Line
Activity 1.18: Vector Equation of a Plane
Activity 1.19: Hyperspace
1.5. Magnitude
Activity 1.20: Length of a Vector
Activity 1.21: Complex Magnitudes
Activity 1.22: Scaling Vectors
1.6. Direction
Activity 1.23: Direction Angles
Activity 1.24: More Direction Angles
Activity 1.25: The Angle Between Vectors
Activity 1.26: Properties of the Dot Product
Activity 1.27: The Dot Product, Revisited
Activity 1.28: The Triangle Inequality, Part One
Activity 1.29: Cauchy-Bunyakovsky-Schwarz
Activity 1.30: The Triangle Inequality, Part Two
Activity 1.31: Cosine Similarity
Activity 1.32: Search Engines
1.7. Unit and Orthogonal Vectors
Activity 1.33: Unit Vectors
Activity 1.34: More About the Dot Product
Activity 1.35: Orthogonal Vectors
2. Systems of Linear Equations
2.1. Standard Form
Activity 2.1: Standard Form
Activity 2.2: The Coefficient Matrix
2.2. Solving Systems
Activity 2.3: Elementary Row Operations
Activity 2.4: Row Echelon Form
Activity 2.5: Row Echelon Form by Fang Cheng Shu
Activity 2.6: Reduced Row Echelon Form by Fang Cheng Shu
2.3. Coefficient Matrices
Activity 2.7: Coefficient Matrices
Activity 2.8: Homogeneous and Inhomogeneous Systems
2.4. Free and Basic Variables
Activity 2.9: Free and Basic Variables
Activity 2.10: Integer Solutions
Activity 2.11: Rows of 0s
Activity 2.12: Rank
2.5. Computational Considerations
Activity 2.13: Roundoff Errors
2.6. Applications of Linear Algebra
Activity 2.14: Finding Orthogonal Vectors
Activity 2.15: Bezout’s Algorithm
Activity 2.16: The Hundred Fowls Problem
Activity 2.17: Shadows
3. Transformations
3.1. Geometric Transformations
Activity 3.1: Geometric Transformations
Activity 3.2: More Rotations
3.2. Vector Transformations
Activity 3.3: Transformations of Vectors
Activity 3.4: More Vector Transformations
3.3. The Transformation Matrix
Activity 3.5: Embeddings
Activity 3.6: More Shadows
3.4. Domain, Codomain, and Range
Activity 3.7: Domain and Codomain
Activity 3.8: Finding the Range, Part One
Activity 3.9: Finding the Range, Part Two
3.5. Discrete Time Models
Activity 3.10: The Rabbit Problem
Activity 3.11: Leslie Models
Activity 3.12: Stochastic Matrices
Activity 3.13: Steady State Vectors
Activity 3.14: How to Lose a Billion Dollars
3.6. Linear Transformations
Activity 3.15: Functions
Activity 3.16: Linear Transformations and Matrices
Activity 3.17: Matrices and Linear Transformations
3.7. Transformation Arithmetic
Activity 3.18: The Identity Matrix
Activity 3.19: Composition of Transformations
Activity 3.20: Inverse Transformations
Activity 3.21: Preserving Linearity
3.8. Cryptography
Activity 3.22: Transposition Ciphers
Activity 3.23: The Hill Cipher
Activity 3.24: More Hills
4. Matrix Algebra
4.1. Scalar Multiplication
Activity 4.1: Scalar Multiplication of a Matrix
Activity 4.2: Equivalent Definitions: Scalar Multiplication
4.2. Matrix Addition
Activity 4.3: Addition of Matrices
Activity 4.4: Equivalent Definitions: Matrix Addition
4.3. Matrix Multiplication
Activity 4.5: Product of Matrices
Activity 4.6: Equivalent Definitions: Matrix Multiplication
Activity 4.7: The Game of Matrix Products
Activity 4.8: Powers of a Matrix and Fast Powering
Activity 4.9: Graphs and Matrices
Activity 4.10: Properties of Matrix Arithmetic
4.4. Elementary Matrices
Activity 4.11: Elementary Matrices
4.5. More Transformations
Activity 4.12: Matrix Multiplication and Transformation
Activity 4.13: Properties of the Transpose
Activity 4.14: The Transpose of a Product, Part One
Activity 4.15: More Transposes
Activity 4.16: Symmetric Matrices
Activity 4.17: Matrices and Rotations
4.6. Matrix Inverses
Activity 4.18: Left Inverses
Activity 4.19: Right Inverses
Activity 4.20: Inverse Matrices
Activity 4.21: Finding the Inverse of a Matrix
Activity 4.22: Double Wide Matrices
Activity 4.23: More Inverses
Activity 4.24: Inverses of Products, Transposes, and Inverses
4.7. Complex Matrices
Activity 4.25: Complex Matrices
Activity 4.26: Hermitian Matrices
5. Vector Spaces
5.1. Vector Spaces
Activity 5.1: Only So Many Symbols
Activity 5.2: Vector Spaces and Subspaces
Activity 5.3: Vector Spaces and the Range
5.2. Kernels and Null Spaces
Activity 5.4: Null Space
Activity 5.5: Properties of the Nullspace
5.3. Span
Activity 5.6: The Ballad of East and West
Activity 5.7: Coordinates
Activity 5.8: Column Space
Activity 5.9: Coordinates
Activity 5.10: Spanning Set
5.4. Linear Independence and Dependence
Activity 5.11: Dependence
Activity 5.12: Steps Towards Independence
Activity 5.13: Gaining Independence
Activity 5.14: Dimension
Activity 5.15: A Basis Exchange
Activity 5.16: Transformation Basis
Activity 5.17: Nothing Counts
5.5. Change of Basis
Activity 5.18: Good Basis, Bad Basis
Activity 5.19: Change of Basis
Activity 5.20: Rotations in R3
5.6. Orthogonal Bases
Activity 5.21: Distance Formulas
Activity 5.22: Orthogonal Bases
5.7. Normed Vector Spaces
Activity 5.23: Another Norm
Activity 5.24: The Secret Life of Norms
Activity 5.25: Complex Norms
Activity 5.26: Even More Norms
5.8. Inner Product Spaces
Activity 5.27: Properties of the Inner Product
Activity 5.28: Inner Products
Activity 5.29: Complexities of the Dot Product
Activity 5.30: More Inner Products
Activity 5.31: Induced Norms
Activity 5.32: Orthogonal Functions
5.9. Applications
Activity 5.33: Dot Products and Frequency Vectors
Activity 5.34: Color Images
Activity 5.35: Lattices
Activity 5.36: More Lattices
Activity 5.37: Lattice Cryptography
Activity 5.38: Quasiorthogonal Basis
5.10. Least Squares
Activity 5.39: Predictions and Observations
Activity 5.40: Squared Deviations
Activity 5.41: Close Approximations
Activity 5.42: Minimizing
Activity 5.43: Least Squares
Activity 5.44: Best Fit Curves
Activity 5.45: “You Might Also Like . . . ”
6. Determinants
6.1. Linear Equations
Activity 6.1: Solving Systems of Equations
6.2. Transformations
Activity 6.2: Transformation of Areas
Activity 6.3: Orientation
Activity 6.4: More Orientation
6.3. Inverses
Activity 6.5: The Inverse of a Matrix
6.4. The Determinant
Activity 6.6: Determinants for Nonsquare Matrices?
Activity 6.7: Algebraic Properties of the Determinant
Activity 6.8: More Algebraic Properties of the Determinant
Activity 6.9: Geometry and the Determinant
Activity 6.10: Switching Rows and Columns
Activity 6.11: Multilinearity of the Determinant
6.5. A Formula for the Determinant
Activity 6.12: Determinant Properties
Activity 6.13: The Determinant of a Diagonal Matrix
Activity 6.14: Determinants of Triangular Matrices
Activity 6.15: Determinant of a 3 3 Matrix
Activity 6.16: Cofactors
Activity 6.17: Cofactor Expansion
Activity 6.18: The Cofactor Checkerboard
6.6. The Determinant Formula
Activity 6.19: Finding Determinants
Activity 6.20: Uniqueness of the Determinant
Activity 6.21: Finding Determinants: Cross Products
6.7. More Properties of the Determinant
Activity 6.22: The Laplace Expansion
Activity 6.23: Determinant of Triangular Matrices
Activity 6.24: More Determinants, More Transformations
Activity 6.25: Determinants of Diagonal and Triangular Matrices
Activity 6.26: More Elementary Matrices
Activity 6.27: Determinants and Rank
Activity 6.28: Determinants and Inverses
Activity 6.29: The Determinant of a Product
Activity 6.30: Determinants and Inverses, Continued
6.8. More Computations of the Determinant
Activity 6.31: Computing the Determinant, Part One
Activity 6.32: Finding Determinants by Row Reduction
Activity 6.33: The LU-Approach to Determinants
6.9. Use(lesses) of the Determinant
Activity 6.34: Cramer’s Rule
Activity 6.35: When to Use Cramer’s Rule
Activity 6.36: The Inverse of a 2 2 Matrix
Activity 6.37: The Adjoint Method
Activity 6.38: When to Use the Adjoint Method
6.10. Uses of the Determinant
Activity 6.39: More Transformations
Activity 6.40: Custom Made Determinants
Activity 6.41: Bad Basis From Good
Activity 6.42: Function Spaces
6.11. Permutations
Activity 6.43: Permutations of Matrices
Activity 6.44: Permutations and the Laplace Expansion
Activity 6.45: Signs of Permutations
Activity 6.46: Properties of Permutations
Activity 6.47: The Permutation Definition of the Determinant
7. Eigenvalues and Eigenvectors
7.1. More Transformations
Activity 7.1: Scaling
Activity 7.2: Stretching
7.2. The Eigenproblem
Activity 7.3: Eigenvectors
Activity 7.4: Properties of Eigenvalues and Eigenvectors
Activity 7.5: Solving the Eigenproblem
Activity 7.6: Finding Eigenvectors
Activity 7.7: Independence of Eigenvectors
7.3. Finding Eigenvalues: Numerical Methods
Activity 7.8: Finding Eigenvalues Numerically
Activity 7.9: Numerical Methods: To the Breaking Point
Activity 7.10: Complex Eigenvalues
7.4. Eigenvalues and Eigenvectors for a 2 x 2 Matrix
Activity 7.11: Finding Eigenvectors
7.5. The Characteristic Equation
Activity 7.12: The Characteristic Equation
Activity 7.13: Eigenvalues and the Characteristic Equation
Activity 7.14: Complex Eigenvalues and Eigenvectors
Activity 7.15: Hermitian Matrices
Activity 7.16: Solving Polynomial Equations
7.6. Stochastic Matrices
Activity 7.17: Eigenvalues and Stochastic Matrices
7.7. A Determinant-Free Approach
Activity 7.18: More Equations for Eigenvalues
Activity 7.19: Higher Dimensional Matrices
Activity 7.20: The Minimal Polynomial
Activity 7.21: Seedling Vectors
7.8. Generalized Eigenvalues
Activity 7.22: Defective Matrices
Activity 7.23: Generalized Eigenvectors
Activity 7.24: Independence of Generalized Eigenvectors
Activity 7.25: Finding Generalized Eigenvectors
Activity 7.26: The Trace
Activity 7.27: Eigenvalues for n x n matrices
7.9. Symmetric Matrices
Activity 7.28: Symmetric Matrices
Activity 7.29: Eigenvalues of Symmetric Matrices
Activity 7.30: Eigenvalues of Symmetric Matrices, Continued
Activity 7.31: Can Symmetric Matrices Be Defective?
Activity 7.32: Positive Definite Matrices
7.10. Graphs
Activity 7.33: More Graphs
Activity 7.34: Centrality Measures
8. Decomposition
8.1. LU-Decomposition
Activity 8.1: Row Reduction, Revisited
Activity 8.2: More Row Reduction
Activity 8.3: Required Row Interchanges
8.2. QR-Decomposition
Activity 8.4: Decomposition Using Gram-Schmidt
8.3. Eigendecompositions
Activity 8.5: Eigendecomposition
Activity 8.6: Diagonalizable Matrices
Activity 8.7: Eigendecompositions With Defective Matrices
Activity 8.8: The Jordan Normal Form
8.4. Singular Value Decomposition
Activity 8.9: More Transformations
Activity 8.10: Stretching and Compressing
Activity 8.11: Singular Value Decomposition
Activity 8.12: More Symmetric Matrices
Activity 8.13: Choices and Ambiguities
Activity 8.14: Sign Ambiguity
Activity 8.15: Singular Value Decomposition
Activity 8.16: Compressing Matrices
9. Extras
9.1. Properties of Polynomials
Activity 9.1: Properties of Polynomials
9.2. Complex Numbers
Activity 9.2: Complex Numbers
Activity 9.3: Complex Arithmetic
Activity 9.4: Conjugates and Polynomials
Activity 9.5: The Complex Plane
9.3. Mod-N Arithmetic
Activity 9.6: Introduction to Mod n Arithmetic
Activity 9.7: Arithmetic mod N
Activity 9.8: Multiplication and Powers Mod N
Activity 9.9: Division mod N
9.4. Polar Coordinates
Activity 9.10: Polar Coordinates
Bibliography
Index