25 lessons across 4 weeks, 66 verified past-year questions. Free preview below — sign in for the full interactive lessons, AI tutor, and mock exams.
Week 1
- What is Machine Learning — Defines machine learning as a higher level of abstraction beyond manual labor and programming, explains when to use it, and illustrates with case studies in face detection, weather prediction, and other applications.
- Data, Models, and Learning Algorithms — Defines the core building blocks of machine learning: data as collections of vectors, models as mathematical simplifications of reality (predictive and probabilistic), and learning algorithms as the tools that convert data into models.
- Supervised Learning and Regression — Introduces the supervised learning paradigm as curve-fitting over labeled data pairs, formalizes regression as predicting real-valued outputs using linear models evaluated by squared loss, and illustrates model selection by comparing candidate models.
- Classification Learning — Formalizes classification as predicting discrete labels using linear separators evaluated by 0-1 loss, and introduces the critical distinction between training, validation, and test data to avoid overfitting.
- Dimensionality Reduction — Introduces dimensionality reduction as an unsupervised learning task that learns an encoder-decoder pair to compress high-dimensional data into a lower-dimensional representation while minimizing reconstruction error.
- Density Estimation — Introduces density estimation as an unsupervised learning task that learns a probabilistic model scoring how likely each data point is, optimized by minimizing negative log likelihood, with applications such as generative tweet models.
Week 2
- Sets, Logic, and Convergence — Recaps the basic language of analysis: sets, metric spaces and balls, logical quantifiers with De Morgan's laws, and convergence of sequences — the vocabulary every later definition is written in.
- Vector Spaces, Dot Products, and Orthogonality — Recaps vector spaces, the dot product and norm, and orthogonality of vectors — the geometric toolkit used throughout the course.
- Functions and Their Visualization — Defines functions and their graphs, and builds visual intuition for one- and two-dimensional functions via surface plots, contour plots, and heat maps.
- Continuity and Differentiability — Defines continuity and differentiability for univariate functions using sequence-based limits, illustrates discontinuous and non-differentiable examples, and interprets the derivative as the slope of the tangent line.
- Linear Approximations and Derivatives — Develops the linear (first-order) approximation of a univariate function from the derivative, derives the product and chain rules through this lens, and applies approximations to compute tangent lines, classic limits, and numerical estimates.
- Critical Points: Maxima, Minima, and Saddle Points — Defines critical points where the derivative vanishes and classifies them as local maxima, minima, or saddle points, motivating their central role in machine learning optimization.
- Geometry of High-Dimensional Space: Lines and Hyperplanes — Establishes the geometry of ℝ^d by defining lines and hyperplanes, distinguishing points from vectors, and building the spatial intuition needed for multivariate calculus.
Week 3
- Four Fundamental Subspaces — Defines and computes the column space, null space, row space, and left null space of a matrix, and establishes how their dimensions relate via rank and nullity.
- Orthogonal Vectors and Subspaces — Defines vector length, orthogonality, and orthonormal vectors, then characterizes which of the four fundamental subspaces are mutually orthogonal.
- Projection onto a Line — Motivates projections via inconsistent linear systems, derives the scalar projection and projection matrix for a line, proves Cauchy-Schwarz as a corollary, and establishes properties P = Pᵀ and P² = P.
- Least Squares and Projection onto a Subspace — Generalizes projection to a subspace by deriving the normal equations AᵀAx̂ = Aᵀb, constructs the projection matrix P = A(AᵀA)⁻¹Aᵀ, and proves its symmetry and idempotence.
- Ordinary Least Squares: Worked Example — Solves a concrete linear regression (curve-fitting) problem end-to-end using the normal equations, computes the best-fit line, and verifies that the residual vector is orthogonal to the column space.
Week 4
- Linear Regression — Formulates the least-squares regression problem as minimizing a sum-of-squares loss and solves it via calculus to obtain the normal equations, connecting the result to maximum likelihood estimation under a Gaussian noise model.
- Polynomial Regression and Ridge Regression — Extends linear regression to polynomial models by transforming input features, and introduces ridge regression as a regularized variant that ensures invertibility and controls overfitting.
- Eigenvalues and Eigenvectors — Defines eigenvalues and eigenvectors via the equation Ax = λx, motivates them through ordinary differential equations, and establishes a systematic procedure for finding them using the characteristic polynomial.
- Diagonalization of a Matrix — Defines diagonalizability in terms of having n linearly independent eigenvectors, proves that eigenvectors corresponding to distinct eigenvalues are linearly independent, and demonstrates the decomposition A = SΛS⁻¹ by example.
- Solving the Fibonacci Sequence using Diagonalization — Applies matrix diagonalization to express the Fibonacci recurrence as a linear system, derives a closed-form approximation for the k-th Fibonacci number using eigenvalues and eigenvectors.
- Orthogonally Diagonalizable Matrices and the Spectral Theorem — States the spectral theorem for real symmetric matrices — that they have real eigenvalues and are orthogonally diagonalizable via A = QΛQᵀ — and demonstrates the decomposition on a 2×2 example.
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