*Linear Algebra* provides us with a unified language for talking
both about systems of simultaneous linear equations and about linear
transformations of space (rotations, reflections, etc). It is the
foundation for almost all mathematics you will study hereafter,
because:

- it is very well-understood: you will see that we have very simple algorithms for solving simultaneous linear equations;
- to study more complicated nonlinear objects for which such methods are lacking, you often try to approximate them by linear ones. For example, calculus is the art of approximating curves and surfaces by straight lines or planes tangent to them.

- how matrices allow us to encode geometric transformations of space as finite arrays of numbers;
- how to encode systems of simultaneous equations in matrix form;
- how to solve matrix equations using row reduction to echelon form (Gaussian elimination);
- how to compute the inverse of a matrix;
- the determinant of a matrix: what it means geometrically and how to compute it;
- eigenvalues and eigenvectors of matrices, with applications to
differential equations, the geometry of ellipsoids, and the
asymptotic behaviour of the Fibonacci sequence.

Because of the 2020 coronavirus pandemic, I am recording my lectures for this module (which runs in April/May 2020). Below you will find links to all the videos, together with HTML transcripts for each video. In the week-by-week plan you will be able to see what work you need to do each week to keep up with the course.

I'll assume you've seen basic trigonometry and calculus, but very little else.

Weekly assessments you can upload to Moodle (with the usual deadlines). The end-of module test will be an open book test running over 48 hours, and there will be no "final exam" this year.

Each page below contains:

- an embedded video (usually lasting 10-20 minutes),
- a set of notes for the video,

- Week 1 (Worksheet 1)
*(approx 2 hours of video)*- 01. Matrices
*(19.05)* - 02. Matrices: examples
*(21.16)* - 03. Bigger matrices
*(24.01)* - 04. Matrix multiplication, 1
*(11.14)* - 05. Matrix multiplication, 2
*(9.14)* - 06. Matrix multiplication, 3
*(11.43)* - 07. Index notation
*(10.09)* - 08. Other operations
*(12.49)*

- 01. Matrices
- Week 2 (Worksheet 2)
*(approx 1 hour 50 mins of video)*- 09. Dot products
*(9.47)* - 10. Dot product 2: transposition
*(9.09)* - 11. Dot product 3: orthogonal matrices
*(15.14)* - 12. Rotations
*(17.41)* - 13. Simultaneous equations and row operations
*(18.53)* - 14. Echelon form
*(10.31)* - 15. Reduced echelon form
*(14.16)* - 16. Echelon examples
*(14.17)*

- 09. Dot products
- Week 3 (Worksheet 3)
*(approx 2.5 hours of video)*- 17. Geometric viewpoint on simultaneous equations, 1
*(11.07)* - 18. Geometric viewpoint on simultaneous equations, 2
*(15.30)* - 19. Subspaces
*(16.47)* - 20. Inverses
*(14.13)* - 21. Inverses: examples
*(18.47)* - 22. Elementary matrices, 1
*(13.41)* - 23. Elementary matrices, 2
*(12.25)* - 24. Determinants
*(22.35)* - 25. Properties of determinants
*(18.55)* - 26. Determinants: examples
*(9.47)*

- 17. Geometric viewpoint on simultaneous equations, 1
- Week 4 (Worksheet 4)
*(approx 2 hours of video)*- 27. Further properties of determinants
*(15.04)* - 28. Cofactor expansion
*(22.16)* - 29. Geometry of determinants, 1
*(9.11)* - 30. Geometry of determinants, 2
*(13.11)* - 31. Eigenvectors and eigenvalues
*(13.02)* - 32. Finding eigenvalues
*(13.23)* - 33. Eigenexamples
*(18.58)* - 34. Eigenspaces
*(12.39)*

- 27. Further properties of determinants
- Week 5
*(approx 2.5 hours of video)*- 35. Eigenapplications: differential equations
*(17.59)* - 36. Eigenapplications: ellipsoids
*(25.11)* - 37. Eigenapplications: dynamics
*(19.45)* - 38. Linear maps
*(16.14)* - 39. Kernels
*(21.04)* - 40. Images
*(22.01)* - 41. Vector spaces
*(29.35)*

- 35. Eigenapplications: differential equations