> For the complete documentation index, see [llms.txt](https://tony-ng-1.gitbook.io/matrices-and-linear-algebra-fundamentals/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://tony-ng-1.gitbook.io/matrices-and-linear-algebra-fundamentals/linear-algebra.md).

# Introduction to Linear Algebra

## Preface

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

{% embed url="<https://www.youtube.com/watch?v=J7DzL2_Na80&t=192s>" %}

## 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**.&#x20;

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**?&#x20;

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.&#x20;

#### **More on the next page ⏭**