Editing Projection (linear algebra) (section)

===Finding projection with an inner product===
Let <math>V</math> be a vector space (in this case a plane) spanned by orthogonal vectors <math>\mathbf u_1, \mathbf u_2, \dots, \mathbf u_p</math>. Let <math>y</math> be a vector. One can define a projection of <math>\mathbf y</math> onto <math>V</math> as
<math display="block"> \operatorname{proj}_V \mathbf y = \frac{\mathbf y \cdot \mathbf u^i}{\mathbf u^i \cdot \mathbf u^i } \mathbf u^i </math>
where repeated indices are summed over ([[Einstein notation|Einstein sum notation]]). The vector <math>\mathbf y</math> can be written as an orthogonal sum such that <math>\mathbf y = \operatorname{proj}_V \mathbf y + \mathbf z</math>. <math>\operatorname{proj}_V \mathbf y</math> is sometimes denoted as <math>\hat{\mathbf y}</math>. There is a theorem in linear algebra that states that this <math>\mathbf z</math> is the smallest distance (the ''[[orthogonal distance]]'') from <math>\mathbf y</math> to <math>V</math> and is commonly used in areas such as [[machine learning]].

[[File:Ortho projection.svg|thumb|''y'' is being projected onto the vector space ''V''.]]