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An orthogonal projection is a projection for which the range ... This formula can be generalized to orthogonal projections on a subspace of arbitrary dimension.
In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace of a vector space equipped with a bilinear form is the set of all vectors in that are orthogonal to every vector in .
Conversely, every projection operator of rank defines a subspace := as its image. Since the rank of an orthogonal projection operator equals its trace , we can identify the Grassmann manifold G r ( k , V ) {\displaystyle \mathbf {Gr} (k,V)} with the set of rank k {\displaystyle k} orthogonal projection operators P {\displaystyle P} :
The vector projection (also known as the vector component or vector resolution) of a vector a on (or onto) a nonzero vector b is the orthogonal projection of a onto a straight line parallel to b. The projection of a onto b is often written as proj b a {\displaystyle \operatorname {proj} _{\mathbf {b} }\mathbf {a} } or a ∥ b .
Hilbert projection theorem — For every vector in a Hilbert space and every nonempty closed convex , there exists a unique vector for which ‖ ‖ is equal to := ‖ ‖.. If the closed subset is also a vector subspace of then this minimizer is the unique element in such that is orthogonal to .
The Gram–Schmidt process takes a finite, linearly independent set of vectors = {, …,} for k ≤ n and generates an orthogonal set ′ = {, …,} that spans the same -dimensional subspace of as . The method is named after Jørgen Pedersen Gram and Erhard Schmidt , but Pierre-Simon Laplace had been familiar with it before Gram and Schmidt. [ 1 ]
In linear algebra, orthogonalization is the process of finding a set of orthogonal vectors that span a particular subspace.Formally, starting with a linearly independent set of vectors {v 1, ... , v k} in an inner product space (most commonly the Euclidean space R n), orthogonalization results in a set of orthogonal vectors {u 1, ... , u k} that generate the same subspace as the vectors v 1 ...
The orthogonal complement of a subspace is the space of all vectors that are orthogonal to every vector in the subspace. In a three-dimensional Euclidean vector space, the orthogonal complement of a line through the origin is the plane through the origin perpendicular to it, and vice versa.