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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.
The orthogonal complement with respect to B of an ideal is again an ideal. [ 3 ] If a given Lie algebra g {\displaystyle {\mathfrak {g}}} is a direct sum of its ideals I 1 ,..., I n , then the Killing form of g {\displaystyle {\mathfrak {g}}} is the direct sum of the Killing forms of the individual summands.
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.
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
Fredholm's theorem in linear algebra is as follows: if M is a matrix, then the orthogonal complement of the row space of M is the null space of M: () = . Similarly, the orthogonal complement of the column space of M is the null space of the adjoint:
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 ...
Hasse diagram of a complemented lattice. A point p and a line l of the Fano plane are complements if and only if p does not lie on l.. In the mathematical discipline of order theory, a complemented lattice is a bounded lattice (with least element 0 and greatest element 1), in which every element a has a complement, i.e. an element b satisfying a ∨ b = 1 and a ∧ b = 0.
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