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Orthogonality, Vector orthogonal to a subset, Orthogonality of two subsets, Orthogonal complement of a vector, Orthogonal complement of a subspace

Orthogonality of two vectors. Two vectors are said to be orthogonal if their inner product is zero.

Vector orthogonal to a subset. A vector x in a vector space V is said to be orthogonal to some subset G in V if it is orthogonal to every vector in the subset G. The subset is called an orthogonal subset of vector x.

Orthogonality of two subsets. Two subsets F and G of a vector space are said to be orthogonal if every vector x in F is orthogonal to every vector y in G.

Orthogonal complement of a vector. The orthogonal complement of a vector x of a vector space is the set of all vectors of the space which are orthogonal to x. Example: The orthogonal complement of a vector in three dimensional space is the set of all vectors perpendicular to the given vector.

Orthogonal complement of a subset. The orthogonal complement of a subset S of a vector space is the set of all vectors of the space which are orthogonal to every vector of S.

Orthogonal complement of a subspace. The orthogonal complement of a subspace S of a vector space is the set T of all vectors of the space which are orthogonal to every vector of S. This set T consisting of the totality of all vectors orthogonal to S is also a subspace. The two subspaces S and T are said to be orthogonal to each other. Subspaces S and T have in common only one element 0. Example: In three dimensional space subspace S could be some plane passing through the origin and T a line perpendicular to the plane.