Linear Algebra

Vectors as arrows, lists and functions; the eight axioms of a vector space and why they matter.

Which subsets are themselves vector spaces, how span builds them, and why a basis is the minimal description.

Assumes: Vectors and Vector Spaces

Testing dependence by elimination, the rank–nullity theorem, and reading rank off a matrix.

Assumes: Subspaces, Span and Basis

Matrices as linear maps: multiplication four different ways, transpose, trace and block structure.

Diagonal, triangular, symmetric, orthogonal, projection, idempotent, nilpotent and partitioned matrices, and the properties each guarantees.

Assumes: Matrices and Their Operations

Existence and uniqueness of solutions, consistency, and the geometry of under- and over-determined systems.

Assumes: Matrices and Their Operations

Row reduction to echelon and reduced row echelon form, pivoting, and operation counts.

Assumes: Systems of Linear Equations

Factorising A into lower and upper triangular parts, partial pivoting, and why LU beats repeated elimination.

Assumes: Gaussian Elimination

Determinants as signed volume: cofactor expansion, elimination-based computation, and the product rule.

Assumes: Gaussian Elimination

Invertibility conditions, Gauss–Jordan inversion, adjugate formula, and the Moore–Penrose pseudoinverse.

Assumes: Determinants

Column space, null space, row space and left null space, and the orthogonality relations that connect them.

Assumes: Linear Independence, Rank and Nullity

Inner products, orthogonal complements, and the projection matrix as the idempotent operator onto a subspace.

Assumes: The Four Fundamental Subspaces

Turning any basis orthonormal, the numerical instability of classical Gram–Schmidt, and QR for least squares.

Assumes: Orthogonality and Projections

Deriving the normal equations three ways — calculus, geometry and projection — the foundation of linear regression.

Assumes: Orthogonality and Projections

The characteristic polynomial, geometric vs algebraic multiplicity, and what eigenvectors mean geometrically.

Assumes: Determinants

When a matrix is diagonalisable, similarity transforms, matrix powers, and the spectral theorem for symmetric matrices.

Assumes: Eigenvalues and Eigenvectors

Writing quadratic forms as xᵀAx, classifying definiteness via eigenvalues and leading minors, and the link to convexity.

Assumes: Diagonalisation and Similarity

Every matrix factors as UΣVᵀ: derivation, geometric reading, low-rank approximation and the Eckart–Young theorem.

Assumes: Diagonalisation and Similarity · The Four Fundamental Subspaces

Vector and matrix norms, equivalence relations between them, and why conditioning decides numerical stability.

Assumes: Singular Value Decomposition

Translating every operation in this module into NumPy, plus the broadcasting and dtype traps that cause silent bugs.

Assumes: Matrices and Their Operations