Linear Algebra • Calculus • Algebra • Algorithm Exercises: from Sorting to Strategic Agents • Supervised Learning

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. 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.

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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. 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.

In progress

• 1. Hello World - Some Short Introductory Coding Exercises; Converting Between Binary, Decimal, and Hexadecimal; Recursive Sequences.
• 2. Searching - Brute Force Search with Linear-Encoding Cryptography; Estimating Roots via Bisection Search and Newton-Rhapson Method.
• 3. Sorting - Selection, Bubble, Insertion, and Counting Sort; Merge Sort and Quicksort.
• 4. Applications of Arrays - Basic Matrix Arithmetic; Reduced Row Echelon Form and Applications to Matrix Arithmetic; Euler Estimation and SIR Model; K-Means Clustering; Simplex Method.
• 5. Graphs - Breadth-First and Depth-First Traversals; Distance and Shortest Paths in Unweighted Graphs; Dijkstra's Algorithm for Distance and Shortest Paths in Weighted Graphs.
• 6. Game Trees - Canonical and Reduced Game Trees for Tic-Tac-Toe; Minimax Strategy and Alpha-Beta Pruning; Pruned Game Trees and Heuristics for Connect Four.

In progress

• 1. The Pseudoinverse - Linear, Polynomial, and Multiple Linear Regression via the Pseudoinverse; Regressing a Linear Combination of Nonlinear Functions via the Pseudoinverse; Power, Exponential, and Logistic Regression via the Pseudoinverse; Overfitting, Underfitting, Cross-Validation, and the Bias-Variance Tradeoff.
• 2. Gradient Descent - Single-Variable Gradient Descent; Multivariable Gradient Descent; Regression via Gradient Descent.
• 3. Multiple Variables - Multiple Regression and Interaction Terms; Principal Component Analysis; Lasso, Ridge, and General Lp Regularization.
• 4. Classifiers - K-Nearest Neighbors; Naive Bayes; Decision Trees, Random Forests, and Boosting; Neural Networks.