Introduction to Linear Algebra

Preface

This may be another good place to remind you to watch his videos before we kick-start Linear Algebra.

System of Linear Equations

As involuntary students of the 21st century, we have all been through some form of algebra in our primary/secondary school (high-school for Americans) 👲, (...to only perform spreadsheets for the rest of our lives 🤡).

Those probably look something a lot like:

$$3x=6$$

I am pretty sure you solved that near instantaneously ⚡, as there is only one equation and thus there is no need to find the solution that simultaneously solves for all equations. This is also otherwise an example of a 1x1 problem, of 1 equation with 1 unknown.

Let's break into a little sweat and see a 2x2 problem then.

$$2x - y = 0 \\ -x + y = 3$$

Now, by simply adding only 1 dimension, we wouldn't be able to see the solution as directly as before (at least for me 👨). Could you possibly imagine 3 dimensions, 4 dimensions, or even 9 dimensions?

We probably could not, but the computer can. However, how do we best communicate these higher-ordered problems in a standardized structure? The answer could be through what mathematics would call a linear system.

In this system,

1. Variables (unknowns) are on the LHS

2. Constants (integers) are on the RHS

3. To use zeroes for the absence of any variables

With this, we are able to express all linear equations with matrices, as would all computer systems.

More on the next page ⏭