Obsidian-Vault/AI Linear Algebra/Orthogonal Projections.md

Orthogonal Projections are a type of projection that maps one vector onto another: ! In this image the result for the projection of x onto y is x_p in order to derive this we can define x_p = cy where c is a mysterious scalar as x_p is always in the same direction as Y and we can define x_o = x-x_p = x-cy and because x_p is orthogonal we can define it as an Inner Product %%Inner Products%% like this: ⟨x − 𝑐y, y⟩ = 0 then to solve for c like this: $$⟨x − 𝑐y, y⟩ = 0 $$⟨x, y⟩-𝑐⟨y, y⟩ = 0 ⟨x, y⟩ = c⟨y, y⟩ \frac{⟨x, y⟩}{⟨y, y⟩} = 𝑐 so after all that we get c=\frac{⟨x, y⟩}{⟨y, y⟩} from this we can derive x_p by doing this: x_p = \frac{⟨x, y⟩}{⟨y, y⟩}y because x_p = cy and c=\frac{⟨x, y⟩}{⟨y, y⟩}.