Engineering Analysis/Projections

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Projection[edit]

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}^Tv

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[edit]

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

Distance between v and W[edit]

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[edit]

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.