Algebra • Calculus • Linear Algebra

I write textbooks for fun as a way to consolidate and clarify my quantitative intuition. The goal is to provide deep intuition for the core concepts and connections, along with plenty of practice exercises, while remaining as concise as possible.

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• 1. Linear Equations and Systems - Solving Linear Equations; Slope-Intercept Form; Point-Slope Form; Standard Form; Linear Systems.
• 2. Quadratic Equations - Standard Form; Factoring; Quadratic Formula; Completing the Square; Vertex Form; Quadratic Systems.
• 3. Inequalities - Linear Inequalities in the Number Line; Linear Inequalities in the Plane; Quadratic Inequalities; Systems of Inequalities.
• 4. Polynomials - Standard Form and End Behavior; Zeros; Rational Roots and Synthetic Division; Sketching Graphs.
• 5. Rational Functions - Polynomial Long Division; Horizontal Asymptotes; Vertical Asymptotes; Graphing with Horizontal and Vertical Asymptotes; Graphing with Slant and Polynomial Asymptotes.
• 6. Non-Polynomial Functions - Radical Functions; Exponential and Logarithmic Functions; Absolute Value; Trigonometric Functions; Piecewise Functions.
• 7. Transformations of Functions - Shifts; Rescalings; Reflections; Inverse Functions; Compositions.

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• 1. Limits and Derivatives - Evaluating Limits; Limits by Logarithms, Squeeze Theorem, and Euler's Consant; Derivatives and the Difference Quotient; Power Rule; Chain Rule; Properties of Derivatives; Derivatives of Non-Polynomial Functions; Finding Local Extrema; Differentials and Approximation; L'Hôpital's Rule.
• 2. Integrals - Antiderivatives; Finding Area; Substitution; Integration by Parts; Improper Integrals.
• 3. Differential Equations - Separation of Variables; Slope Fields and Euler Approximation; Substitution; Characteristic Polynomial; Undetermined Coefficients; Integrating Factors; Variation of Parameters.
• 4. Series - Geometric Series; Tests for Convergence; Taylor Series; Manipulating Taylor Series; Solving Differential Equations with Taylor Series.

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• 1. Vectors - N-Dimensional Space; Dot Product and Cross Product; Lines and Planes; Span, Subspaces, and Reduction; Elimination as Vector Reduction.
• 2. Volume - N-Dimensional Volume Formula; Volume as the Determinant of a Square Linear System; Shearing, Cramer’s Rule, and Volume by Reduction; Higher-Order Variation of Parameters.
• 3. Matrices - Linear Systems as Transformations of Vectors by Matrices; Matrix Multiplication; Rescaling, Shearing, and the Determinant; Inverse Matrices.
• 4. Eigenspace - Eigenvalues, Eigenvectors, and Diagonalization; Recursive Sequence Formulas via Diagonalization; Generalized Eigenvectors and Jordan Form; Matrix Exponential and Systems of Linear Differential Equations.

In progress

• 1. Basic - Some Short Introductory Coding Exercises; Converting Between Binary, Decimal, and Hexadecimal; Recursive Sequences; Selection, Bubble, Insertion, and Counting Sort; Brute Force Search with Linear-Encoding Cryptography; Estimating Roots via Bisection Search and Newton-Rhapson Method; Matrix Arithmetic and Recursive Determinant; Euler Estimation and SIR Model; Breadth-First and Depth-First Traversal; Distance and Shortest Paths in Unweighted Directed Graphs.
• 2. Intermediate - Merge Sort; Reduced Row Echelon Form, Matrix Inverse, and Efficient Determinant; Hodgkin-Huxley Model; Learning Simplex Method via Algebra; Implementing Simplex Method via Array Operations; Distance and Shortest Paths in Weighted Directed Graphs; Canonical and Reduced Tic-Tac-Toe Game Trees; K-Means Clustering; Algorithmic Complexity.
• 3. Advanced - QuickSort; Solving Magic Squares and Sudoku via Search Pruning; Minimax Strategy; Alpha-Beta Pruning; Q-Learning on Blackjack.

In progress

• 1. The Pseudoinverse - Linear, Polynomial, and Multiple Regression via the Pseudoinverse; Regressing a Linear Combination of Nonlinear Functions via the Pseudoinverse; Power, Exponential, and Logistic Regression via the Pseudoinverse; Interaction Terms; Principal Component Analysis; Feature Selection.
• 2. Validation - Overfitting, Underfitting, Cross-Validation, and the Bias-Variance Tradeoff; Class Imbalance, Sampling, Confusion Matrix, and Receiver Operating Characteristic.
• 3. Gradient Descent - Single-Variable Gradient Descent; Multivariable Gradient Descent; Regression via Gradient Descent; Regularization.
• 4. Metaheuristics - Hill Climbing, Tabu Search, and Simulated Annealing; Swarm-Based Optimization; Evolutionary Algorithms; Combining Metaheuristics and Gradient Descent.
• 5. More Models - K-Nearest Neighbors and Normalization; Decision Trees; Random Forests, Adaptive Boosting, and Gradient Boosting; Support Vector Machines; Naive Bayes and Kernel Density Estimation.

To be written!

• 1. Neural Networks - Architecture, Initialization, and Data Normalization in Neural Networks; Computing Weight Gradients via Chain Rule; Computing Weight Gradients via Path Enumeration; Computing Weight Gradients via Backpropagation; Evolving Weights in Neural Networks; Data Augmentation and Adversarial Examples; Dropout and Batch Normalization.
• 2. Convolutional Neural Networks - Convolutional and Pooling Layers; ...
• 3. Applications of Convolutional Neural Networks - Object Localization; Deep Dream; Neural Style Transfer; Deepfakes; Blondie24; ...
• 4. Recurrent Neural Networks - Long Short-Term Memory Units; Backpropagtion Through Time; ...
• 5. Applications of Recurrent Neural Networks - Text Summarization; Image Captioning; Translation; Chatbots; ...