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EspañolQUOTIENT SPACE (LINEAR ALGEBRA)
In linear algebra, the 'quotient' of a vector space ''V'' by a subspace ''N'' is a vector space obtained by "collapsing" ''N'' to zero. The space obtained is called a 'quotient space' and is denoted ''V''/''N'' (read ''V'' mod ''N'').
Formally, the construction is as follows. Let ''V'' be a vector space over a field ''K'', and let ''N'' be a subspace of ''V''. We define an equivalence relation ~ on ''V'' by stating that ''x'' ~ ''y'' if ''x'' − ''y'' ∈ ''N''. That is, ''x'' is related to ''y'' if one can be obtained from the other by adding an element of ''N''. The equivalence class of ''x'' is often denoted
:[''x''] = ''x'' + ''N''
since it is given by
:[''x''] = {''x'' + ''n'' : ''n'' ∈ ''N''}.
The quotient space ''V''/''N'' is then defined as ''V''/~, the set of all equivalence classes over ''V'' by ~. Scalar multiplication and addition are defined on the equivalence classes by
★ α[''x''] = [α''x''] for all α ∈ ''K'', and
★ [''x''] + [''y''] = [''x''+''y''].
It is not hard to check that these operations are well-defined (i.e. do not depend on the choice of representative). These operations turn the quotient space ''V''/''N'' into a vector space over ''K''.
This simplest example is to take a quotient of 'R'''n''. Let ''m'' ≤ ''n'' and let 'R'''m'' be the subspace spanned by the first ''m'' standard basis vectors. Two vectors of 'R'''n'' are then seen to be equivalent if and only if they are identical in the last ''n''−''m'' coordinates. The quotient space 'R'''n''/ 'R'''m'' is isomorphic to 'R'''n''−''m'' in an obvious manner.
More generally, if ''V'' is written as an (internal) direct sum of subspaces ''U'' and ''W'':
:
then the quotient space ''V''/''U'' is naturally isomorphic to ''W''.
If ''U'' is a subspace of ''V'', the 'codimension' of ''U'' in ''V'' is defined to be the dimension of ''V''/''U''. If ''V'' is finite-dimensional, this is just the difference in the dimensions of ''V'' and ''U'':
:
There is a natural epimorphism from ''V'' to the quotient space ''V''/''U'' given by sending ''x'' to its equivalence class [''x'']. The kernel (or nullspace) of this epimorphism is the subspace ''U''. This relationship is neatly summarized by the short exact sequence
:
Let ''T'' : ''V'' → ''W'' be a linear operator. The kernel of ''T'', denoted ker(''T''), is the set of all ''x'' ∈ ''V'' such that ''Tx'' = 0. The kernel is a subspace of ''V''. The first isomorphism theorem of linear algebra says that the quotient space ''V''/ker(''T'') is isomorphic to the image of ''V'' in ''W''. An immediate corollary, for finite-dimensional spaces, is the rank-nullity theorem: the dimension of ''V'' is equal to the dimension of the kernel (the ''nullity'' of ''T'') plus the dimension of the image (the ''rank'' of ''T'').
The cokernel of a linear operator ''T'' : ''V'' → ''W'' is defined to be the quotient space ''W''/im(''T'').
If ''X'' is a Banach space and ''M'' is a closed subspace of ''X'', then the quotient ''X''/''M'' is again a Banach space. The quotient space is already endowed with a vector space structure by the construction of the previous section. We define a norm on ''X''/''M'' by
:
The quotient space ''X''/''M'' is complete with respect to the norm, so it is a Banach space.
Let ''C''[0,1] denote the Banach space of continuous real-valued functions on the interval [0,1]. Denote the subspace of all functions ''f'' ∈ ''C''[0,1] with ''f''(0) = 0 by ''M''. Then the equivalence class of some function ''g'' is determined by its value at 0, and the quotient space ''C''[0,1] / ''M'' is isomorphic to 'R'.
If ''X'' is a Hilbert space, then the quotient space ''X''/''M'' is isomorphic to the orthogonal complement of ''M''.
★ quotient set
★ quotient group
★ quotient module
★ quotient space (in topology)
| Contents |
| Definition |
| Examples and properties |
| Quotient of a Banach space by a subspace |
| Examples |
| See also |
Definition
Formally, the construction is as follows. Let ''V'' be a vector space over a field ''K'', and let ''N'' be a subspace of ''V''. We define an equivalence relation ~ on ''V'' by stating that ''x'' ~ ''y'' if ''x'' − ''y'' ∈ ''N''. That is, ''x'' is related to ''y'' if one can be obtained from the other by adding an element of ''N''. The equivalence class of ''x'' is often denoted
:[''x''] = ''x'' + ''N''
since it is given by
:[''x''] = {''x'' + ''n'' : ''n'' ∈ ''N''}.
The quotient space ''V''/''N'' is then defined as ''V''/~, the set of all equivalence classes over ''V'' by ~. Scalar multiplication and addition are defined on the equivalence classes by
★ α[''x''] = [α''x''] for all α ∈ ''K'', and
★ [''x''] + [''y''] = [''x''+''y''].
It is not hard to check that these operations are well-defined (i.e. do not depend on the choice of representative). These operations turn the quotient space ''V''/''N'' into a vector space over ''K''.
Examples and properties
This simplest example is to take a quotient of 'R'''n''. Let ''m'' ≤ ''n'' and let 'R'''m'' be the subspace spanned by the first ''m'' standard basis vectors. Two vectors of 'R'''n'' are then seen to be equivalent if and only if they are identical in the last ''n''−''m'' coordinates. The quotient space 'R'''n''/ 'R'''m'' is isomorphic to 'R'''n''−''m'' in an obvious manner.
More generally, if ''V'' is written as an (internal) direct sum of subspaces ''U'' and ''W'':
:
then the quotient space ''V''/''U'' is naturally isomorphic to ''W''.
If ''U'' is a subspace of ''V'', the 'codimension' of ''U'' in ''V'' is defined to be the dimension of ''V''/''U''. If ''V'' is finite-dimensional, this is just the difference in the dimensions of ''V'' and ''U'':
:
There is a natural epimorphism from ''V'' to the quotient space ''V''/''U'' given by sending ''x'' to its equivalence class [''x'']. The kernel (or nullspace) of this epimorphism is the subspace ''U''. This relationship is neatly summarized by the short exact sequence
:
Let ''T'' : ''V'' → ''W'' be a linear operator. The kernel of ''T'', denoted ker(''T''), is the set of all ''x'' ∈ ''V'' such that ''Tx'' = 0. The kernel is a subspace of ''V''. The first isomorphism theorem of linear algebra says that the quotient space ''V''/ker(''T'') is isomorphic to the image of ''V'' in ''W''. An immediate corollary, for finite-dimensional spaces, is the rank-nullity theorem: the dimension of ''V'' is equal to the dimension of the kernel (the ''nullity'' of ''T'') plus the dimension of the image (the ''rank'' of ''T'').
The cokernel of a linear operator ''T'' : ''V'' → ''W'' is defined to be the quotient space ''W''/im(''T'').
Quotient of a Banach space by a subspace
If ''X'' is a Banach space and ''M'' is a closed subspace of ''X'', then the quotient ''X''/''M'' is again a Banach space. The quotient space is already endowed with a vector space structure by the construction of the previous section. We define a norm on ''X''/''M'' by
:
The quotient space ''X''/''M'' is complete with respect to the norm, so it is a Banach space.
Examples
Let ''C''[0,1] denote the Banach space of continuous real-valued functions on the interval [0,1]. Denote the subspace of all functions ''f'' ∈ ''C''[0,1] with ''f''(0) = 0 by ''M''. Then the equivalence class of some function ''g'' is determined by its value at 0, and the quotient space ''C''[0,1] / ''M'' is isomorphic to 'R'.
If ''X'' is a Hilbert space, then the quotient space ''X''/''M'' is isomorphic to the orthogonal complement of ''M''.
See also
★ quotient set
★ quotient group
★ quotient module
★ quotient space (in topology)
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