Quotient space (linear algebra)
The factor space (also quotient space ) is a term from linear algebra, a branch of mathematics. He is the one vector space which arises as the image of a parallel projection along a subspace. The elements of the factor space are equivalence classes.
Definition
It is a vector space over a field and a vector subspace of. By fixing
Is defined on an equivalence relation.
The vectors and are equivalent if they differ by a vector of. Other words, if those are straight line through the points and is parallel and equivalent.
The equivalence class of a point
Vividly to the "parallel" affine subspace through. The equivalence classes are also referred to as cosets; This term comes from the group theory.
The factor of space by the set of all equivalence classes and is denoted by:
He forms a vector space if the vector space operations are defined as representative:
For and.
These operations are well defined, ie on the choice of representatives independent.
Properties
- There is a canonical surjective linear map
- Is a complement of in, that is, is the direct sum of and so is the restriction of an isomorphism., There is no canonical way to be construed as subspace.
- Is finite, then this results in the following relation for the dimensions:
- The dual space can be identified with those of the linear shapes that are identical to 0.
- The homomorphism theorem says that a linear map is an isomorphism
Application in Functional Analysis
Many normed spaces arise in the following way: Let be a real or complex vector space and is a semi-norm on. Then a subspace is. The factor space is then the norm is a normed vector space.
General: Let be a topological vector space which is not Hausdorff. Then can be analogous to above define a subspace. The factor space with the quotient topology is a Hausdorff topological vector space shear.
Examples
Abstract
The rooms and thus the Sobolev spaces are factor spaces.
Concrete
Be a vector space and a one-dimensional subspace of.
Is an equivalence class of the factor space.
Clearly, any straight line which is parallel to the bisecting line of the first quadrant, an equivalence class: