Engineering Analysis/Projections

Projection
The projection of a vector $$v \in V$$ onto the vector space $$W \in V$$ is the minimum distance between v and the space W. In other words, we need to minimize the distance between vector v, and an arbitrary vector $$w \in W$$:


 * $$\|w - v\|^2 = \|\hat{W}\hat{a} - v\|^2$$


 * $$\frac{\partial \|\hat{W} \hat{a} - v\|^2}{\partial \hat{a}} = \frac{\partial \langle \hat{W}\hat{a} - v, \hat{W}\hat{a} - v\rangle }{\partial \hat{a}} = 0$$


 * $$\hat{a} = (\hat{W}^T\hat{W})^{-1}\hat{W}^Tv$$

For every vector $$v \in V$$ there exists a vector $$w \in W$$ called the projection of v onto W such that  = 0, where p is an arbitrary element of W.

Orthogonal Complement

 * $$w^\perp = {x \in V: \langle x, y \rangle = 0, \forall y \in W}$$

Distance between v and W
The distance between $$v \in V$$ and the space W is given as the minimum distance between v and an arbitrary $$w \in W$$:


 * $$\frac{\partial d(v, w)}{\partial \hat{a}} = \frac{\partial\|v - \hat{W}\hat{a}\|}{\partial \hat{a}} = 0$$

Intersections
Given two vector spaces V and W, what is the overlapping area between the two? We define an arbitrary vector z that is a component of both V, and W:


 * $$z = \hat{V} \hat{a} = \hat{W} \hat{b}$$


 * $$\hat{V} \hat{a} - \hat{W} \hat{b} = 0$$


 * $$\begin{bmatrix}\hat{a} \\ \hat{b}\end{bmatrix}= \mathcal{N}([\hat{v} - \hat{W}])$$

Where N is the nullspace.