So, turns out, the answer I searched was correct... It was on the next page. Well, lesson learned, I'll finish a section from now on if possible instead of using a time measured learning system.

## The answer to my previous post (Day 10)

### let's say I have these vectors

### To know if they're linearly dependent:

#### Equals:

#### Sooo

If we use the RREF it'll be:

And since every element of the column is a pivot column, there's no trivial solution making every single column a unique column which means :

and these vectors are linearly independent from one another.

## Generating Set and Basis

Finally, from a long time ago I didn't know what a basis was and it's going to be discussed today!

### Consider the vector space:

#### and a set of vectors

If every vector set of the vector is a part of the vector space, meaning:

and can be expressed as a linear combination of A (Yes, that absolute horrendous letter is an A but using \mathscr). Then A is a generating set of V

The set of all linear combination of the vectors A is called the span of A.

#### Consider a vector space:

If there's no smaller set, then the last generating set is called a minimal\basis.

### One more example

#### Let

#### and

#### Then,

- B is a basis of V
- B is a minimal generating set
- B is a maximal linearly independent set of vectors in V, i.e. adding any other vectors to this set will make it linearly dependent
- Every vector

and every linear combinations are unique, i.e. with:

with

## Thoughts

Honestly, today's section is pretty great to read, since today is focused on answering the previous chapter's question / confusion and finally learn what the hell is a basis, it almost feels like eating a dessert after a buffet, not too difficult but enough since it completes a few gaps in the previous sections.

### A little TL;DR:

- A generating set is a subset of the vector space that can recreate the vector subspace with linear combination
- A generating set can be linearly dependent, since it's not a smaller chunk of the vector space, but can be an entirely different vector all together, but can recreate the subspace
- A basis on the other hand is a linearly independent vector subspace, since this is the smallest known generating set that the vector space can create. So removing even a single vector from this set will remove the ability to recreate a portion of the vector space.

## 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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