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Mathematics

The visual math spine under machine learning and algorithms: vectors, matrices, gradients.

The shared spine under both algorithms and machine learning. Build the math I can reason from without looking it up: sets and functions, vectors and matrices, derivatives and gradients. Visual first, so that every embedding and every training step later stops being magic.

Written by Claude Opus 4.8 19 lessons, 7 modules updated Jul 2026

Module 0 Foundations

start here

Sets, relations, and functions

The vocabulary every other idea is built from

Exponentials and logarithms

Growth, decay, and the inverse that tames them

Module 1 Linear Algebra

the language of ML

Vectors and vector spaces

Arrows and lists: one object, two faces

Dot product and cosine similarity

Measuring alignment, length, and similarity

Matrices and matrix multiplication

Linear maps and the operation ML runs on

Module 2 Advanced Linear Algebra

the geometry deepens

Linear transformations

Matrices as functions that move space

Eigenvectors and eigenvalues

The directions a transformation only scales

Module 3 Calculus for Learning

how models improve

Derivatives and the chain rule

Slope, sensitivity, and composing change

Partial derivatives and gradients

The gradient: which way is uphill

Module 4 Discrete Math and Proof

reasoning rigorously

Logic, quantifiers, and the anatomy of a proof

The grammar of rigorous mathematical argument

Proof by induction and recursion

Proving a statement for every natural number

Summations, series, and asymptotic sums

Adding many terms and estimating the total

Counting and combinatorics

How many ways: permutations and combinations

Module 5 Probability and Statistics

uncertainty and data

Descriptive statistics and the Gaussian

Summarizing data: mean, variance, the bell curve

Probability, random variables, and distributions

Formalizing chance and its shapes

Conditional probability and Bayes

Updating beliefs when evidence arrives

Maximum likelihood estimation

Choosing parameters that best explain the data

Information theory: entropy and cross-entropy

Measuring surprise and the loss that trains LLMs

Module 6 Numerical Foundations

how machines compute

Floating point and numerical stability

Why computers get arithmetic slightly wrong

Reference shelf

Glossary

Every term in this spine, crisply defined and cross-linked.

Formula sheet

The core formulas of the course, on one page to revisit.

How to use this course

Do the widgets. The quizzes and flashcards are not decoration: trying to recall before you reveal is what makes it stick.
Space it out. Come back a day later and answer them cold.
Read critically. An LLM wrote these, so verify anything before you rely on it.