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 $w\in W$:

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

$\frac{\partial \Vert \hat{W}\hat{a}-v{\Vert }^2}{\partial \hat{a}}=\frac{\partial ⟨\hat{W}\hat{a}-v,\hat{W}\hat{a}-v⟩}{\partial \hat{a}}=0$

[Projection onto space W]

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

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

Orthogonal Complement

$w^{\perp }=x\in V:⟨x,y⟩=0,\mathrm{\forall }y\in W$

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 $w\in W$:

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

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:

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

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

$\left[\begin{array}{c}\hat{a}\\ \hat{b}\end{array}\right]=𝒩([\hat{v}-\hat{W}])$

Where N is the nullspace.