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Concept
Orthogonal complement
Summary
In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace W of a vector space V equipped with a bilinear form B is the set W⊥ of all vectors in V that are orthogonal to every vector in W. Informally, it is called the perp, short for perpendicular complement. It is a subspace of V.
Let be the vector space equipped with the usual dot product (thus making it an inner product space), and let with
then its orthogonal complement can also be defined as being
The fact that every column vector in is orthogonal to every column vector in can be checked by direct computation. The fact that the spans of these vectors are orthogonal then follows by bilinearity of the dot product. Finally, the fact that these spaces are orthogonal complements follows from the dimension relationships given below.
Let be a vector space over a field equipped with a bilinear form We define to be left-orthogonal to , and to be right-orthogonal to when For a subset of define the left orthogonal complement to be
There is a corresponding definition of right orthogonal complement. For a reflexive bilinear form, where implies for all and in the left and right complements coincide. This will be the case if is a symmetric or an alternating form.
The definition extends to a bilinear form on a free module over a commutative ring, and to a sesquilinear form extended to include any free module over a commutative ring with conjugation.
An orthogonal complement is a subspace of ;
If then ;
The radical of is a subspace of every orthogonal complement;
If is non-degenerate and is finite-dimensional, then
If are subspaces of a finite-dimensional space and then
Orthogonal projection
This section considers orthogonal complements in an inner product space
Two vectors and are called if which happens if and only if for all scalars
If is any subset of an inner product space then its is the vector subspace
which is always a closed subset of that satisfies and also and
If is a vector subspace of an inner product space then
If is a closed vector subspace of a Hilbert space then
where is called the of into and and it indicates that is a complemented subspace of with complement
The orthogonal complement is always closed in the metric topology.
Official source
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Ontological neighbourhood