Orthogonalkomplement

Orthogonalkomplement, also known as orthogonal complement, is a concept in linear algebra that refers to the set of all vectors in a vector space that are orthogonal to a given subspace. Orthogonality in this context means that the dot product of any vector in the subspace and any vector in its orthogonal complement is zero.

The orthogonal complement of a subspace V in a vector space U is denoted as V⊥. If V is a subspace of U, then V⊥ is also a subspace of U. The dimension of V⊥ is equal to the codimension of V in U, which is the difference between the dimension of U and the dimension of V.

The orthogonal complement is a fundamental concept in linear algebra and has numerous applications in mathematics, physics, and engineering. It is used in the study of vector spaces, linear transformations, and inner product spaces. The orthogonal complement is also used in the construction of orthogonal bases and in the solution of systems of linear equations.