Orthogonal complement

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".
Example.
In the case that "W" is the subspace of formula_1 (with the usual dot product) spanned by the rows of the next matrix,
formula_2
its orthogonal complement "W"⊥ is spanned by the three row-vectors of
formula_3
The fact that every vector on the first list is orthogonal to every vector on the second list 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.
General bilinear forms.
Let formula_4 be a vector space over a field formula_5 equipped with a bilinear form formula_6. We define formula_7 to be left-orthogonal to formula_8, and formula_8 to be right-orthogonal to formula_7, when formula_11. For a subset formula_12 of formula_4 we define the left orthogonal complement formula_14 to be
There is a corresponding definition of right orthogonal complement. For a reflexive bilinear form, where formula_11 implies formula_17 for all formula_7 and formula_8 in formula_4, the left and right complements coincide. This will be the case if formula_6 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.
Inner product spaces.
This section considers orthogonal complements in inner product spaces.
Properties.
The orthogonal complement is always closed in the metric topology. In finite-dimensional spaces, that is merely an instance of the fact that all subspaces of a vector space are closed. In infinite-dimensional Hilbert spaces, some subspaces are not closed, but all orthogonal complements are closed. In such spaces, the orthogonal complement of the orthogonal complement of formula_12 is the closure of formula_12, i.e.,
Some other useful properties that always hold are the following. Let formula_36 be a Hilbert space and let formula_37 and formula_38 be its linear subspaces. Then:
The orthogonal complement generalizes to the annihilator, and gives a Galois connection on subsets of the inner product space, with associated closure operator the topological closure of the span.
Finite dimensions.
For a finite-dimensional inner product space of dimension "n", the orthogonal complement of a "k"-dimensional subspace is an -dimensional subspace, and the double orthogonal complement is the original subspace:
If "A" is an matrix, where , , and refer to the row space, column space, and null space of "A" (respectively), we have
Banach spaces.
There is a natural analog of this notion in general Banach spaces. In this case one defines the orthogonal complement of "W" to be a subspace of the dual of "V" defined similarly as the annihilator
It is always a closed subspace of "V"∗. There is also an analog of the double complement property. "W"⊥⊥ is now a subspace of "V"∗∗ (which is not identical to "V"). However, the reflexive spaces have a natural isomorphism "i" between "V" and "V"∗∗. In this case we have
This is a rather straightforward consequence of the Hahn–Banach theorem.
Applications.
In special relativity the orthogonal complement is used to determine the simultaneous hyperplane at a point of a world line. The bilinear form η used in Minkowski space determines a pseudo-Euclidean space of events. The origin and all events on the light cone are self-orthogonal. When a time event and a space event evaluate to zero under the bilinear form, then they are hyperbolic-orthogonal. This terminology stems from the use of two conjugate hyperbolas in the pseudo-Euclidean plane: conjugate diameters of these hyperbolas are hyperbolic-orthogonal.
References.
4. Orthogonal complement ; Minute 9.00 in the Youtube Video