> For the complete documentation index, see [llms.txt](https://stephanosterburg.gitbook.io/scrapbook/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://stephanosterburg.gitbook.io/scrapbook/math/math-for-ml/linear-algebra/wk2.md).

# Wk2

## Modulus & Inner Product

The length of a vector, also called its size, and the dot product of a vector, also called it's inner scalar or projection product.

![Size of Vector](https://3501392451-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LLZ89zzVxrdnG1RG6CA%2F-Li_dVF7RmjuItNzZMwX%2F-LiO_GIf1RqxND8ShZ8J%2FScreen%20Shot%202019-06-27%20at%208.36.35%20AM.png?alt=media\&token=71e5c239-7d63-4b8a-be56-110d20af2977)

![](https://3501392451-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LLZ89zzVxrdnG1RG6CA%2F-Li_dVF7RmjuItNzZMwX%2F-LiOc7RB0ciIjuMZXn1J%2FScreen%20Shot%202019-06-27%20at%208.48.43%20AM.png?alt=media\&token=9826bf4e-1566-4252-b4a1-996eed0ac102)

## Cosine & dot product

![](https://3501392451-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LLZ89zzVxrdnG1RG6CA%2F-Li_dVF7RmjuItNzZMwX%2F-LiOe0ETfzx7a6Nar6gW%2FScreen%20Shot%202019-06-27%20at%208.56.58%20AM.png?alt=media\&token=5a8c55cf-1adc-4b7b-bd2f-8df7be2b300e)

![](https://3501392451-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LLZ89zzVxrdnG1RG6CA%2F-Li_dVF7RmjuItNzZMwX%2F-LiOgWmZcx_MgcYTeA_b%2FScreen%20Shot%202019-06-27%20at%209.08.24%20AM.png?alt=media\&token=586ecf43-45cb-4e0f-bcbc-cb3c38494569)

## Changing Basis

![](https://3501392451-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LLZ89zzVxrdnG1RG6CA%2F-Li_dVF7RmjuItNzZMwX%2F-LiTxohEmTHPLpQ0pe5n%2FScreen%20Shot%202019-06-28%20at%209.15.35%20AM.png?alt=media\&token=eea5e196-ec8f-4f66-8e44-e246e785964a)

## **Basis, vector space, and linear independence**

![](https://3501392451-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LLZ89zzVxrdnG1RG6CA%2F-Li_dVF7RmjuItNzZMwX%2F-LiTyAqvrJMomcg0pvex%2FScreen%20Shot%202019-06-28%20at%209.43.10%20AM.png?alt=media\&token=d2bef838-22ec-4849-b147-08d3612de1c4)

![](https://3501392451-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LLZ89zzVxrdnG1RG6CA%2F-Li_dVF7RmjuItNzZMwX%2F-LiTyghzJEn59-i2ZR3x%2FScreen%20Shot%202019-06-28%20at%209.45.53%20AM.png?alt=media\&token=271a52d7-bc96-4fa1-8080-a5d4d7737b4a)

Now if we're thinking about a neural network in machine learning that recognizes faces say, maybe I'd want to make some transformation of all the pixels interface into a new basis that describes the nose shape, the skin hue, the distance between the eyes those sorts of things and discard the actual pixel data. So the goal of the learning process of the neural network is going to be to somehow derive a set of basis vectors that extract the most information-rich features of the faces.