Open Semester

Mathematics · 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.

Start with Lecture 1

Weeks 1 to 5 are out. 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.
Start here The Language of Mathematics A short primer on the notation every lecture uses: sets, quantifiers, implications, functions and sums. 12:11 Open the primer

Weeks

L = lecture, R = recitation. Each page has the video, the book section, the problem sheet and a short check.

Week 1

Sheet 1 PDF Solutions PDF
  1. L1 What linear algebra is about 15:43
  2. L2 Adding and scaling vectors 13:33
  3. L3 Linear combinations and span 17:55
  4. L4 Dot product, length and angle 18:04
  5. R1 Recitation 1Problem Sheet 1, worked through 31:36

Week 2

Sheet 2 PDF Solutions PDF
  1. L5 Lines, planes and the cross product 18:04
  2. L6 Fields 17:31
  3. L7 Systems of linear equations 16:57
  4. L8 Gaussian elimination 17:20
  5. R2 Recitation 2Problem Sheet 2, worked through 30:19

Week 3

Sheet 3 PDF Solutions PDF
  1. L9 Reduced row echelon form 12:24
  2. L10 Solution sets 15:29
  3. L11 Linear independence 18:36
  4. L12 The matrix-vector product 19:42
  5. R3 Recitation 3Problem Sheet 3, worked through 30:21

Week 4

Sheet 4 PDF Solutions PDF
  1. L13 Linear transformations of Rn 17:05
  2. L14 Matrix multiplication 12:41
  3. L15 Matrix algebra 16:36
  4. L16 Inverse matrices 17:55
  5. R4 Recitation 4Problem Sheet 4, worked through 30:17

Week 5

Sheet 5 PDF Solutions PDF
  1. L17 The invertible matrix theorem 15:57
  2. L18 LU factorisation 15:51
  3. L19 Subspaces of Rn 12:54
  4. L20 Basis and dimension in Rn 16:50
  5. R5 Recitation 5Problem Sheet 5, worked through 30:25

Week 6

Coming soon
  1. L21Vector spaces
  2. L22Subspaces, sums and intersections
  3. L23Basis and dimension in general
  4. L24Direct sums
  5. R6Recitation 6

Week 7

Coming soon
  1. L25Coordinates
  2. L26Row space and rank
  3. L27The four fundamental subspaces
  4. L28Change of basis
  5. R7Recitation 7
Practice midtermWeeks 1 to 7, with full solutionsComing soon

Week 8

Coming soon
  1. L29Linear maps
  2. L30The rank–nullity theorem
  3. L31Isomorphisms and the space of maps
  4. L32The matrix of a linear map
  5. R8Recitation 8

Week 9

Coming soon
  1. L33Determinant as signed volume
  2. L34The determinant defined
  3. L35Computing determinants
  4. L36Determinants and invertibility
  5. R9Recitation 9

Week 10

Coming soon
  1. L37Eigenvalues and eigenvectors
  2. L38The characteristic polynomial
  3. L39Diagonalisation
  4. L40Eigenvalues in action
  5. R10Recitation 10

Week 11

Coming soon
  1. L41Complex eigenvalues
  2. L42Invariant subspaces
  3. L43Polynomials of matrices
  4. L44The minimal polynomial
  5. R11Recitation 11

Week 12

Coming soon
  1. L45Inner product spaces
  2. L46Orthogonality and projections
  3. L47Gram–Schmidt and QR
  4. L48Least squares
  5. R12Recitation 12

Week 13

Coming soon
  1. L49The spectral theorem
  2. L50Quadratic forms
  3. L51Singular value decomposition
  4. L52The SVD in action and what comes next
  5. R13Recitation 13
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.

  1. 1 Vectors and the geometry of RnLectures 1 to 5 PDF · 33 pp.
  2. 2 Fields and linear systemsLectures 6 to 11 PDF · 41 pp.
  3. 3 Matrices and linear transformationsLectures 12 to 18 PDF · 56 pp.
  4. 4 Vector spacesLectures 19 to 24 · Sections 4.1 to 4.2 so far PDF · 15 pp.
  5. 5Coordinates, rank and change of basisLectures 25 to 28Coming soon
  6. 6Linear mapsLectures 29 to 32Coming soon
  7. 7DeterminantsLectures 33 to 36Coming soon
  8. 8Eigenvalues and diagonalisationLectures 37 to 40Coming soon
  9. 9Invariant subspaces and the minimal polynomialLectures 41 to 44Coming soon
  10. 10Inner product spacesLectures 45 to 48Coming soon
  11. 11The spectral theorem and the SVDLectures 49 to 52Coming soon
  12. RRulebookEvery definition, theorem and proposition so far, in orderPDF · 21 pp.

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