Engineering Analysis/Projections

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Projection[edit | edit source]

The projection of a vector onto the vector space 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 :

[Projection onto space W]

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

Orthogonal Complement[edit | edit source]

Distance between v and W[edit | edit source]

The distance between and the space W is given as the minimum distance between v and an arbitrary :

Intersections[edit | edit source]

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:

Where N is the nullspace.