Math lessons & courses
32 lessons · 8 learning paths · free, quiz-checked, no signup required
The mathematics that underpins engineering and data work: linear algebra, statistics, and the habits of quantitative reasoning. Each lesson builds the intuition first, then makes it precise.
Learning paths
Convex Optimization: From Gradient Descent to Interior Point Methods
The real divide in optimization is convex against nonconvex, and it decides whether you can prove your answer is best or merely hope so. Start by learning to recognise convexity without touching a Hessian. Derive why the safe step size is one over the smoothness constant, and why the condition number governs everything. Pick up duality and the KKT conditions as your certificate. Then add curvature, and see how a log barrier turned constrained problems into something solvable in polynomial time.
Group Theory and Symmetry
A group is a set with one operation and four rules, and that deliberate poverty is what lets a single theorem cover rotations, shuffles, clock arithmetic and Rubik's cube at once. This path starts from the symmetries of a square, proves Lagrange's theorem and gets Fermat's little theorem for free, shows why no crystal can have five-fold symmetry and what happened when one did, and ends with the group of order 60 that makes a quintic formula impossible.
Discrete Mathematics: Counting, Proof, Graphs, and Congruence
The mathematics of finite structures sits under most of computing, and it is usually absorbed in fragments rather than learned. This path builds four working tools: counting arrangements you could never enumerate, proving a claim for every input instead of the ones you tested, modelling relationships as graphs and reading what the structure permits, and doing arithmetic on a circle. Every result is tied to something engineers actually hit.
Signal Processing: Fourier, Sampling, and Filters
Audio, images, sensors and instruments all rest on the same four ideas, and getting any of them wrong destroys data silently. This path builds the frequency domain and why sinusoids get the job, the sampling rule that cannot be broken and what aliasing does when it is, convolution and the filters built from it, and how to read a spectrum without mistaking the analysis for the signal. Every number is computed, so you can reproduce it.
Calculus for Machine Learning
Derivatives, gradients and the chain rule are the machinery under every training run, and knowing them changes what you can debug. This path builds the derivative as a local linear model, then gradients, Jacobians and Hessians, then the chain rule as a product whose evaluation order decides the cost, and finally automatic differentiation itself. You will finish knowing why a billion-parameter gradient costs one backward pass, why conditioning sets your iteration count, and where autodiff answers the wrong question exactly.
Optimization on Manifolds: When Parameters Live on a Curve
Rotations, subspaces, low-rank matrices and covariances are not vectors in flat space, and treating their constraints as penalties throws away structure that makes the problem easier. This path builds Riemannian optimization from the ground up: tangent spaces and the metric that decides what steepest means, the projection that turns an ordinary gradient into the right one, retractions that let you move along a curved space without solving differential equations, and why these non-convex problems are so often solved to global optimality anyway.
Probability and Statistics for Machine Learning
Build the mathematical foundation every ML practitioner needs: go from sample spaces and distributions to Bayesian inference and hypothesis testing. By the end you will be able to choose the right distribution for any modelling problem, derive maximum likelihood estimators, reason about uncertainty the Bayesian way, and correctly interpret p-values and confidence intervals.
Linear Algebra for Engineers
Master the mathematical backbone of machine learning, signal processing, and scientific computing. By the end you will decompose any matrix into its fundamental subspaces, compute eigenvalues and eigenvectors, apply the SVD for low-rank approximation and compression, and solve least-squares problems — all with geometric intuition and NumPy.
All Math lessons
Expectation, Variance, and the CLT
Master the three numbers that summarize any distribution: mean, variance, and standard deviation. Derive linearity of expectation, understand covariance and correlation, then see why the Central Limit Theorem makes the Gaussian unavoidable — with a worked numeric example from scratch.
Estimation and Hypothesis Testing
From raw data to defensible conclusions: derive Maximum Likelihood Estimators for Bernoulli and Gaussian, understand bias-variance in estimation, construct confidence intervals, and learn what p-values actually say — and don't say — including the most common misinterpretation that has corrupted thousands of papers.
Bayesian Inference
Understand what it really means to update beliefs with data. Derive Bayes' theorem from first principles, dissect the roles of prior, likelihood, posterior, and evidence, work through a complete Beta-Binomial conjugate example numerically, and see why the base-rate fallacy trips up even experts.
Vectors, Spans, and Subspaces
Vectors are more than arrows — they're the atoms of every ML model, physics engine, and signal processor alive. Build rock-solid intuition for linear combinations, span, independence, basis, and orthogonality, then verify it all in NumPy.
SVD and Least Squares
When there's no exact solution, project. When data is high-dimensional, compress. The SVD is the Swiss Army knife that does both — and more. Master orthogonal projection, the normal equations, the Singular Value Decomposition, low-rank approximation, and the pseudoinverse.
Matrices as Linear Transformations
A matrix doesn't just hold numbers — it reshapes space. Master the geometric view of matrix-vector multiplication, the four fundamental subspaces, rank, the determinant as a volume-scaling factor, and invertibility — all grounded in NumPy.
Eigenvalues and Eigenvectors
Some vectors only get scaled by a matrix — they don't rotate at all. These eigenvectors reveal the skeleton of a linear transformation. Master the eigen-equation, the characteristic polynomial, diagonalization, and why eigenstructure powers PCA, PageRank, and stability analysis.
Lagrangian Duality: From Primal to Dual
Every constrained optimization problem has a twin. Learn how to build the Lagrangian, derive the dual problem, and use weak duality, strong duality, and the KKT conditions to certify optima — with worked examples from linear programming and SVMs.