Syllabus

Units are written in the order the companion build course needs them, but they are organised here in logical order. Status: written, next, or planned. The plan will change as the course grows.

Foundations

Unit	Topic	Status
F1	Sets, families and intersections	written
F2	Functions in depth: injective, surjective, inverse, composition	planned
F3	Relations, equivalence classes, orders	planned
F4	Proof toolkit: direct, contradiction, contrapositive, induction	planned
F5	Countable and uncountable sets	planned

Linear algebra

Unit	Topic	Status
LA1	Vector spaces	written
LA2	Span	written
LA3	Linear independence	next
LA4	Bases and dimension	planned
LA5	Linear maps and matrices	planned
LA6	Null space, range, rank–nullity	planned
LA7	Systems of linear equations, as questions about spans	planned
LA8	Norms, inner products, orthogonality	planned
LA9	Projections and least squares	planned
LA10	Determinants, as volume	planned
LA11	Eigenvalues and eigenvectors	planned
LA12	Symmetric matrices and the spectral theorem	planned
LA13	Singular value decomposition and low-rank approximation	planned

Calculus and optimisation

Unit	Topic	Status
C1	Limits and continuity	planned
C2	Derivatives, single variable	planned
C3	Partial derivatives and the gradient	planned
C4	Jacobians and the chain rule	planned
C5	Taylor expansion and the Hessian	planned
C6	Convexity	planned
C7	Gradient-based optimisation	planned
C8	Multiple integrals and change of variables	planned

Probability, statistics, information

Unit	Topic	Status
P1	Probability spaces and events (σ-algebras, gently)	planned
P2	Conditional probability and Bayes’ rule	planned
P3	Random variables, expectation, variance	planned
P4	Continuous and joint distributions, the Gaussian	planned
P5	Limit theorems	planned
S1	Estimation: likelihood, MLE, MAP	planned
S2	Uncertainty and tests: confidence intervals, p-values, the bootstrap	planned
I1	Entropy, cross-entropy and KL divergence	planned