This week, we will learn about homogeneous systems. The idea is that if you want to solve a system of linear equations, then it is almost enough to consider the corresponding homogeneous system. You can read more about this in my lecture notes and I will talk about this in the class.

So far, we have learned 4 different languages to talk about the same thing:

1. Systems of linear equations
2. Vector equations (linear combination, spanning etc.)
3. Matrix equations (Ax=b)
4. Linear functions (respecting the two structures)

We are going to spend some time this week to discuss what it means to be "a solution of a homogeneous system" in terms of these languages. Words you will learn: nullspace, kernel, linear independence, linearly independent and more... Words, so many words... But you see, all these concepts are very connected with each other. If you can see the relation between them, the course will become much much clearer.