Course · 13 weeks
Linear Algebra
A full first university course in linear algebra, sized to a 13-week semester: 52 lectures, 13 recitations, an 11-chapter course book, 13 weekly problem sheets with worked solutions, and two practice exams. Geometry first, then rigour: every idea is shown as a picture in motion, then stated precisely, then proved.
Weeks 1 to 5 available: 20 lectures, 5 recitations, 5 problem sheets. New weeks are added as they are finished.
- Who it is for
- Anyone with high-school algebra who wants the real university course, from curious adults to first-year students.
- You need
- Solving equations, functions, coordinates in the plane. No calculus. Proofs are taught from scratch.
- Each week
- Four lectures of about 12 to 20 minutes, one recitation of 30 to 40 minutes, and a problem sheet with full worked solutions.
Weeks
L = lecture, R = recitation. Each lecture page has its video link, book section, problem sheet and contents.
Week 6
- L21Vector spaces
- L22Subspaces, sums and intersections
- L23Basis and dimension in general
- L24Direct sums
- R6Recitation 6Problem Sheet 6, worked through
Week 7
- L25Coordinates
- L26Row space and rank
- L27The four fundamental subspaces
- L28Change of basis
- R7Recitation 7Problem Sheet 7, worked through
Practice midtermWeeks 1 to 7, with full solutionsComing soon
Week 8
- L29Linear maps
- L30The rank–nullity theorem
- L31Isomorphisms and the space of maps
- L32The matrix of a linear map
- R8Recitation 8Problem Sheet 8, worked through
Week 9
- L33Determinant as signed volume
- L34The determinant defined
- L35Computing determinants
- L36Determinants and invertibility
- R9Recitation 9Problem Sheet 9, worked through
Week 10
- L37Eigenvalues and eigenvectors
- L38The characteristic polynomial
- L39Diagonalisation
- L40Eigenvalues in action
- R10Recitation 10Problem Sheet 10, worked through
Week 11
- L41Complex eigenvalues
- L42Invariant subspaces
- L43Polynomials of matrices
- L44The minimal polynomial
- R11Recitation 11Problem Sheet 11, worked through
Week 12
- L45Inner product spaces
- L46Orthogonality and projections
- L47Gram–Schmidt and QR
- L48Least squares
- R12Recitation 12Problem Sheet 12, worked through
Week 13
- L49The spectral theorem
- L50Quadratic forms
- L51Singular value decomposition
- L52The SVD in action and what comes next
- R13Recitation 13Problem Sheet 13, worked through
Practice finalAll weeks, with full solutionsComing soon
Course book
One section per lecture, with full definitions, every proof and extra worked examples. Chapters are published as their weeks are finished.
- 1Vectors and the geometry of RnLectures 1 to 5
- 2Fields and linear systemsLectures 6 to 11
- 3Matrices and linear transformationsLectures 12 to 18
- 4Vector spacesLectures 19 to 24 · Sections 4.1 to 4.2 so far
- 5Coordinates, rank and change of basisLectures 25 to 28
- 6Linear mapsLectures 29 to 32
- 7DeterminantsLectures 33 to 36
- 8Eigenvalues and diagonalisationLectures 37 to 40
- 9Invariant subspaces and the minimal polynomialLectures 41 to 44
- 10Inner product spacesLectures 45 to 48
- 11The spectral theorem and the SVDLectures 49 to 52
- RRulebookEvery definition, theorem and proposition so far, in order