Difference between revisions of "Codimension"
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− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , '''1''' , Addison-Wesley (1974) pp. Chapt.1;2 (Translated from French) {{MR|0354207}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Differentiable and analytic manifolds" , Addison-Wesley (1966) (Translated from French) {{MR|0205211}} {{MR|0205210}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> M. Golubitsky, "Stable mappings and their singularities" , Springer (1973) {{MR|0341518}} {{ZBL|0294.58004}} </TD></TR></table> |
Revision as of 16:56, 15 April 2012
The codimension (or quotient or factor dimension) of a subspace of a vector space
is the dimension of the quotient space
; it is denoted by
, or simply by
, and is equal to the dimension of the orthogonal complement of
in
. One has
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If and
are two subspaces of
of finite codimension, then
and
are also of finite codimension, and
![]() |
The codimension of a submanifold of a differentiable manifold
is the codimension of the tangent subspace
of the tangent space
at
. If
and
are finite-dimensional, then
![]() |
If and
are differentiable manifolds, if
is a submanifold of
and if
is a differentiable mapping transversal to
, then
![]() |
The codimension of an algebraic subvariety (or an analytic subspace) of an algebraic variety (analytic space)
is the difference
![]() |
References
[1] | N. Bourbaki, "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , 1 , Addison-Wesley (1974) pp. Chapt.1;2 (Translated from French) MR0354207 |
[2] | N. Bourbaki, "Elements of mathematics. Differentiable and analytic manifolds" , Addison-Wesley (1966) (Translated from French) MR0205211 MR0205210 |
[3] | M. Golubitsky, "Stable mappings and their singularities" , Springer (1973) MR0341518 Zbl 0294.58004 |
Comments
The codimension of a subspace of a vector space
is equal to the dimension of any complement of
in
, since all complements have the same dimension (as the orthogonal complement).
Codimension. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Codimension&oldid=24399