# Factor space

The **quotient vector space** , also called **quotient space** or **factor space** for short , is a term from linear algebra , a branch of mathematics . It is the vector space that arises as an image of a parallel projection along a sub-vector space . The elements of the quotient vector space are equivalence classes.

## definition

Let it be a vector space over a field and a subspace of . By fixing

- For

is defined on an equivalence relation .

The vectors and are therefore equivalent if they differ by one vector . In other words: If the line through the points and to parallel are is, and equivalent.

The equivalence class of a point is

- ,

clearly shows the "parallel" affine subspace . The equivalence classes are also known as secondary classes (this term comes from group theory ).

The *quotient vector space* from to is the set of all equivalence classes and is denoted by:

- .

It forms a vector space if the vector space operations are defined by proxy:

for and .

These operations are well-defined, i.e. independent of the choice of representatives.

## properties

- There is a canonical surjective linear mapping

- .

- Is a complement of in , i.e. H. is the direct sum of and , the restriction of to is an isomorphism . But there is no canonical possibility to understand it as a subspace of .

- If it is finite-dimensional, then the following relationship results for the dimensions:

- The dual space of can be identified with those linear forms on that are identical on.

- The homomorphism theorem states that a linear mapping has an isomorphism

- between the quotient space from after the kernel from and the image from induced, d. H. the concatenation
- is the same .

## Application in functional analysis

Many normed spaces created in the following way: Be a real or complex vector space and let 1 seminorm on . Then is a subspace of . The quotient space then becomes a normalized vector space with the norm .

More general: Let be a topological vector space that is not Hausdorffian . Then be analogous to above subspace define: . The quotient space with the quotient topology becomes a Hausdorff topological vector space.

## Examples

### abstract

The spaces and thus also the Sobolew spaces are quotient vector spaces.

### Concrete

The vector space and the one-dimensional sub-vector space are given . Then for example

an equivalence class of the quotient space .

Every straight line that is parallel to the bisecting line of the 1st quadrant is clearly an equivalence class:

## See also

## literature

- Gerd Fischer:
*Linear Algebra.*Vieweg-Verlag, ISBN 3-528-97217-3 . - Klaus Jänich:
*Linear Algebra.*Springer textbook, ISBN 3-540-66888-8 .