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 $w\in W$:

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

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

[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

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

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.