## Dear mathmaticians.

I love you math and mathematicians... but why? Today I spent the most time trying to understand and the least amount page read... And so, it has become the meme of the day.

What happened? It all began with:

## Algorithms for Solving System of Linear Equations

Until now, the examples and matrices chosen always has an answer, but if we don't make the assumption that there isn't a solution, we can make an approximation to the solution.

One way is using the approach of linear regression.

if A is a square matrix and is invertible (non-zero determinant)

If A does have a determinant of zero, we are using something called:

### Moore-Penrose Pseudo-inverse

Though we can solve using these methods. There are some disadvantages when it comes to it:

- Requiring many computations for matrix-matrix product
- Requiring the important role of Gaussian elimination, what for?

- Compute determinant
- Compute inverse
- Check for linearly independancy
- Compute the rank of a matrix
- Find basis of a vector space

### Well... Technically

In practice there are many linear equations that can be solved indirectly,

- Stationary iterative method

- Richardson method
- Jacobi method
- Gauss-Seidel method
- Successive over-relaxation method

- Krylov subspace method

- Conjugative gradients
- Generalized minimal residual
- Biconjugate gradients

### Vector Space

A structured space in which vectors live. I know, very helpful explanation.

But before we can learn about vector space, there's something that needs to be established.

## Group

A set of elements and an operation defined on these elements that keep some structure of the set intact.

This is where the fun begins.

### Conditions

Now, to make sure you understand how confused I was, I'll notate the equations first then continue with the explanation.

### Where's the numbers?

Good question. I don't know, even the inverse portion doesn't have any -1 on it. Which I have to say... Impressive. Let's start with the very first thing.

#### G and G

I won't lie, neither KaTeX nor LaTeX have this G and further search leads me to no where. So I assume it's the same thing, which is a group. Here's what the custom-GPT `math`

said

#### So what's a set?

If you're like me, you basically forgot this even existed aside from inside programming languages. It's very similar.

A set is a collection of unique/distinct objects. For example:

{1,2,3,...} is a set of natural numbers

{2,4,6,...} is a set of even numbers

Now you might think that "Oh, that's easy. Let's move on". Then let me ask you this:

Which one of these is a set?

Yup, it's both. What about this?

Still both. Finally, this:

I don't have to say it right? The Fibonacci sequence isn't a set.

#### \bigotimes or ⨂ (This is the KaTeX notation)

\bigotimes or Big'O as the kids say these days is a placeholder for binary operations. Not to be confused with or which also uses ⨂ and I spent so much time thinking it was these.

##### What do I mean by placeholder?

Let me give you an example

Y can be 2, 1/2 and -2. Since Big'O can be an addition, a subtraction, a multiplication. All three answer is correct depending on further information.

Big'O can also be used in matrices or sets with the same concept. Here are the things Big'O can replace:

- Addition
- Subtraction
- Multiplication
- Union
- Intersect

With that out of the way, let's continue with the conditions!

### Closure

Closure means, with x and y being an element of the set G, the result of the operation is also an element from the set G.

That means, if G is a set of natural numbers, with X ⨂ Y being elements of the set G, we can deduce that:

- X and Y > 0
- X > Y
- If X ⨂ Y = Z, Z is an element from the set G.

### Associativity

This is something we've already discussed extensively, so I won't go too long, it just means that the order of the operation doesn't matter.

### Neutral element

Neutral element is an element that when operated with any other value, the result will said `other value`

.

Here are what counts as neutral elements:

### Inverse element

This condition goes hand in hand with the last condition, with the value x there exist y that when operated against x will return a neutral number.

Here's what I mean:

This means that the inverse of x is -3. What's interesting, the author notes that though inverse is often noted with -1 on the top of the element, it doesn't always mean to be divided by said value.

## And that's it for today.

If you think it's pretty short and without any mathematical background you can understand very quickly, you're welcome. I'm kidding, sort of. I took me so long to understand those notations and what it really means to be a set, operation, etc.

Since the author said this part is very important in computer science, I want to make sure I understood as clearly as possible and It'll take a while :D

## P.S.

I'm starting to understand more and more regarding mathematical notations but whenever I feel like I can breeze a few pages, math just slaps me in the face and kick me in the nuts. Feels bad man.

## Acknowledgement

I can't overstate this: I'm truly grateful for this book being open-sourced for everyone. Many people will be able to learn and understand machine learning on a fundamental level. Whether changing careers, demystifying AI, or just learning in general, this book offers immense value even for *fledgling composer* such as myself. So, Marc Peter Deisenroth, A. Aldo Faisal, and Cheng Soon Ong, thank you for this book.

Source:

Deisenroth, M. P., Faisal, A. A., & Ong, C. S. (2020). Mathematics for Machine Learning. Cambridge: Cambridge University Press.

https://mml-book.com

## Top comments (0)