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:

Fundamental theorem on homomorphisms Kolmogorov space
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