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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 .
1. Orthogonal complement: If W is a linear subspace of an inner product space V, then denotes its orthogonal complement, that is, the linear space of the elements of V whose inner products with the elements of W are all zero. 2.
Thus A T x = 0 if and only if x is orthogonal (perpendicular) to each of the column vectors of A. It follows that the left null space (the null space of A T) is the orthogonal complement to the column space of A. For a matrix A, the column space, row space, null space, and left null space are sometimes referred to as the four fundamental subspaces.
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. [ 5 ] Note that the geometric concept of two planes being perpendicular does not correspond to the orthogonal complement, since in three dimensions a pair of vectors, one from ...
In the branch of mathematics called functional analysis, a complemented subspace of a topological vector space, is a vector subspace for which there exists some other vector subspace of , called its (topological) complement in , such that is the direct sum in the category of topological vector spaces.
It follows that x is in the kernel of A, if and only if x is orthogonal (or perpendicular) to each of the row vectors of A (since orthogonality is defined as having a dot product of 0). The row space, or coimage, of a matrix A is the span of the row vectors of A. By the above reasoning, the kernel of A is the orthogonal complement to the row
The vector space of complex-valued class functions of a group has a natural -invariant inner product structure, described in the article Schur orthogonality relations.Maschke's theorem was originally proved for the case of representations over by constructing as the orthogonal complement of under this inner product.
In mathematics, orthogonal functions belong to a function space that is a vector space equipped with a bilinear form. When the function space has an interval as the domain , the bilinear form may be the integral of the product of functions over the interval:
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