## An offer you literally can refuse.

You're not gonna like today, it's going to be really short since I'm only going to cover one section and it's been a long day. Though I'd rather be consistent than anything else, it doesn't negate the fact that this is a poor time management from me. Since I'm not a politician, I won't sugarcoat that this will happen and I can't guarantee this won't happen again but I'll keep moving forward.

## Rank

So, you're stuck in low ELO hell, be it in chess, Valorant or Dota 2, we've all been there but how do you get from low ELO (or dare I say deficient rank) to a full rank? But we're going too far, let's start all the way back.

### What's a rank?

A rank is the number of linearly independent columns of a matrix

equals the number of linearly independent rows. The rank of a is denoted by:

Is it really that simple?

### Of course not.

#### Properties of rank

#### 1. The column rank equals the row rank

#### 2. The columns of

with the dimensions of U equal the rank of A. Later on, this subspace will be called the image/range.

#### 3. The rows of

with the dimensions of W equal the rank of A. A basis of W can be found by applying Gaussian elimination to A.

#### 4. For all

it holds that the linear equation system Ax = b can be solved if and only if the rank of A equals the rank of A|b, where A|b denotes the augmented system.

#### 5. For all

posesses dimensions

Later on we'll call ths subspace the kernel or the null space.

#### 6. A matrix

has full rank if its rank equals the largest possible rank for a matrix of the same dimensions. This means that the rank of a full rank matrix is the lesser of the number of rows and columns. i.e. rk(A) = min(m,n). A matrix is said to be rank deficient if it doesn't have full rank.

## Conclusion.

If you're friend is asking for tips to climb the rank? Just tell them to be a matrix with each column being a pivot column.

Because honestly, a full rank rank is just a basis but in matrices and not vectors.

## 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

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