# 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 ${\ displaystyle V}$ ${\ displaystyle K}$${\ displaystyle U}$${\ displaystyle V}$

${\ displaystyle v_ {1} \ sim v_ {2} \;: \! \ iff v_ {1} -v_ {2} \ in U}$ For ${\ displaystyle v_ {1}, v_ {2} \ in V}$

is defined on an equivalence relation . ${\ displaystyle V}$

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. ${\ displaystyle v_ {1}}$${\ displaystyle v_ {2}}$${\ displaystyle U}$${\ displaystyle v_ {1}}$${\ displaystyle v_ {2}}$${\ displaystyle U}$${\ displaystyle v_ {1}}$${\ displaystyle v_ {2}}$

The equivalence class of a point is ${\ displaystyle v}$

${\ displaystyle [v]: = v + U: = \ {v + u \ mid u \ in U \}}$,

clearly shows the "parallel" affine subspace . The equivalence classes are also known as secondary classes (this term comes from group theory ). ${\ displaystyle U}$${\ displaystyle v}$

The quotient vector space from to is the set of all equivalence classes and is denoted by: ${\ displaystyle V}$${\ displaystyle U}$${\ displaystyle V / U}$

${\ displaystyle V / U: = \ {[v] \ mid v \ in V \}}$.

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

• ${\ displaystyle [v_ {1}] + [v_ {2}] = [v_ {1} + v_ {2}]}$
• ${\ displaystyle \ lambda \ cdot [v] = [\ lambda v]}$

for and . ${\ displaystyle v, v_ {1}, v_ {2} \ in V}$${\ displaystyle \ lambda \ in K}$

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

## properties

${\ displaystyle \ pi \ colon \; V \ to V / U, \; v \ mapsto [v]}$.
• 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 .${\ displaystyle W}$${\ displaystyle U}$${\ displaystyle V}$${\ displaystyle V}$${\ displaystyle U}$${\ displaystyle W}$${\ displaystyle \ pi}$${\ displaystyle W}$${\ displaystyle V / U}$${\ displaystyle V}$
• If it is finite-dimensional, then the following relationship results for the dimensions:${\ displaystyle V}$
${\ displaystyle \ dim U + \ dim V / U = \ dim V}$
• The dual space of can be identified with those linear forms on that are identical on.${\ displaystyle V / U}$${\ displaystyle V}$${\ displaystyle U}$${\ displaystyle 0}$
• The homomorphism theorem states that a linear mapping has an isomorphism${\ displaystyle f \ colon \; V \ to W}$
${\ displaystyle V / (\ ker f) \ to \ mathrm {im} \, f}$
between the quotient space from after the kernel from and the image from induced, d. H. the concatenation ${\ displaystyle V}$${\ displaystyle f}$${\ displaystyle f}$
${\ displaystyle V \ longrightarrow V / (\ ker f) \ longrightarrow \ mathrm {im} \, f \ longrightarrow W}$
is the same .${\ displaystyle f}$

## 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 . ${\ displaystyle V}$${\ displaystyle p}$${\ displaystyle V}$${\ displaystyle U = \ {v \ in V \ mid p (v) = 0 \}}$${\ displaystyle V}$${\ displaystyle V / U}$ ${\ displaystyle [v] \ mapsto p (v)}$

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. ${\ displaystyle V}$${\ displaystyle U = \ {v \ in V \ mid {\ text {Each}} 0 {\ text {-Environment contains}} v \} = {\ overline {\ {0 \}}}}$${\ displaystyle V / U}$

## Examples

### abstract

The spaces and thus also the Sobolew spaces are quotient vector spaces. ${\ displaystyle L ^ {p}}$

### Concrete

The vector space and the one-dimensional sub-vector space are given . Then for example ${\ displaystyle V = \ mathbb {R} ^ {2}}$${\ displaystyle U = \ left \ {\ left. {\ bigl (} {\ begin {smallmatrix} x \\ x \ end {smallmatrix}} {\ bigr)} \ right | x \ in \ mathbb {R} \ right \}}$

${\ displaystyle {\ bigl (} {\ begin {smallmatrix} 42 \\ 12 \ end {smallmatrix}} {\ bigr)} + ​​U: = \ left \ {\ left. {\ bigl (} {\ begin {smallmatrix } 42 \\ 12 \ end {smallmatrix}} {\ bigr)} + ​​u \, \ right | u \ in U \ right \}}$

an equivalence class of the quotient space . ${\ displaystyle V / U}$

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