# Projection Formally, we can define projection as follows. Let: $\omega: X \rightarrow Y$ be a **projection operator** that maps points in a space $X$ to points in a subspace $Y \subset X$. ### Intuition > Generally, a projection is an operation that moves points into a subspace. ### Coordinate Functions are projection operations It may not be immediately apparent, but the coordinate functions themselves are literally projection operations! For example, consider $\mathbb{R}^3$. If we have a point $(1,4,2)$ and we have a function that maps that point to a the first coordinate, $x_1$, then we have: $proj((1,4,2)) = 1$ Which is literally a projection! We have mapped from $\mathbb{R}^3$ to $\mathbb{R}$, where $\mathbb{R}$ is of course a subspace of $\mathbb{R}^3$. ### Projections and information loss There is of course information that is lost when performing a projection. ### Geometric Projection In geometric projection a point will always move to the closest point on the subspace that it is being projected onto (see fantastic video [here](https://youtu.be/wciU07gPqUE?list=PLWhu9osGd2dB9uMG5gKBARmk73oHUUQZS&t=404)). ### Resources The two main definitions that are useful regarding projections are: * [Linear algebra projection](https://en.wikipedia.org/wiki/Projection_(linear_algebra)) * This is highly related to the [Dot Product](Dot%20Product.md) * [Projection of one vector onto another, whiteboard](https://photos.google.com/photo/AF1QipORmIdHiAvIm7SMxWEVxAj-X8kfqjVMjH-EpZxW) * [Projection of one vector onto another, whiteboard 2](https://photos.google.com/photo/AF1QipO4YixJCk0snqKEuz78hNSi86fg_QuX_typ-gZx) * [Explained via Fourier Series](https://youtu.be/MB6XGQWLV04?list=PLMrJAkhIeNNT_Xh3Oy0Y4LTj0Oxo8GqsC&t=527) * Chapter 6, linear algebra and it's applications * [Linear regression as a projection](https://medium.com/@vladimirmikulik/why-linear-regression-is-a-projection-407d89fd9e3a) * Notability: linear regression as projection * Notability: Probability theory, 2.2.1 (betancourt)