# MATH105 Linear Algebra

## Module overview

### What is Linear Algebra about?

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.
We will see:
• 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.

### Video lectures

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.

### Prerequisites

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

### Assessment

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.

## Index of all video lectures

Each page below contains:

• an embedded video (usually lasting 10-20 minutes),
• a set of notes for the video,
The notes are annotated with times (these annotations should look like (8.30)) which indicate whereabouts in the video you can see that section of the notes. I have also indicated the length of the videos below similarly. The total length of all videos is about 10 hours and 53 minutes.