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The codimension (or quotient or factor dimension) of a subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022870/c0228701.png" /> of a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022870/c0228702.png" /> is the dimension of the quotient space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022870/c0228703.png" />; it is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022870/c0228704.png" />, or simply by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022870/c0228705.png" />, and is equal to the dimension of the orthogonal complement of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022870/c0228706.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022870/c0228707.png" />. One has
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The codimension (or quotient or factor dimension) of a subspace $L$ of a vector space $V$ is the dimension of the quotient space $V/L$; it is denoted by $\codim_VL$, or simply by $\codim L$, and is equal to the dimension of the orthogonal complement of $L$ in $V$. One has
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022870/c0228708.png" /></td> </tr></table>
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$$\dim L+\codim L=\dim V.$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022870/c0228709.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022870/c02287010.png" /> are two subspaces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022870/c02287011.png" /> of finite codimension, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022870/c02287012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022870/c02287013.png" /> are also of finite codimension, and
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If $M$ and $N$ are two subspaces of $V$ of finite codimension, then $M\cap N$ and $M+N$ are also of finite codimension, and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022870/c02287014.png" /></td> </tr></table>
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$$\codim(M+N)+\codim(M\cap N)=\codim M+\codim N.$$
  
The codimension of a submanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022870/c02287015.png" /> of a differentiable manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022870/c02287016.png" /> is the codimension of the tangent subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022870/c02287017.png" /> of the tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022870/c02287018.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022870/c02287019.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022870/c02287020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022870/c02287021.png" /> are finite-dimensional, then
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The codimension of a submanifold $N$ of a differentiable manifold $M$ is the codimension of the tangent subspace $T_x(N)$ of the tangent space $T_x(M)$ at $x\in N$. If $M$ and $N$ are finite-dimensional, then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022870/c02287022.png" /></td> </tr></table>
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$$\codim N=\dim M-\dim N.$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022870/c02287023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022870/c02287024.png" /> are differentiable manifolds, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022870/c02287025.png" /> is a submanifold of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022870/c02287026.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022870/c02287027.png" /> is a differentiable mapping transversal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022870/c02287028.png" />, then
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If $M$ and $N$ are differentiable manifolds, if $L$ is a submanifold of $N$ and if $f\colon M\to N$ is a differentiable mapping transversal to $L$, then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022870/c02287029.png" /></td> </tr></table>
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$$\codim f^{-1}(L)=\codim L.$$
  
The codimension of an algebraic subvariety (or an analytic subspace) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022870/c02287030.png" /> of an algebraic variety (analytic space) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022870/c02287031.png" /> is the difference
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The codimension of an algebraic subvariety (or an analytic subspace) $Y$ of an algebraic variety (analytic space) $X$ is the difference
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022870/c02287032.png" /></td> </tr></table>
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$$\codim Y=\dim X-\dim Y.$$
  
 
====References====
 
====References====
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====Comments====
 
====Comments====
The codimension of a subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022870/c02287033.png" /> of a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022870/c02287034.png" /> is equal to the dimension of any complement of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022870/c02287035.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022870/c02287036.png" />, since all complements have the same dimension (as the orthogonal complement).
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The codimension of a subspace $L$ of a vector space $V$ is equal to the dimension of any complement of $L$ in $V$, since all complements have the same dimension (as the orthogonal complement).

Latest revision as of 22:29, 30 November 2018

The codimension (or quotient or factor dimension) of a subspace $L$ of a vector space $V$ is the dimension of the quotient space $V/L$; it is denoted by $\codim_VL$, or simply by $\codim L$, and is equal to the dimension of the orthogonal complement of $L$ in $V$. One has

$$\dim L+\codim L=\dim V.$$

If $M$ and $N$ are two subspaces of $V$ of finite codimension, then $M\cap N$ and $M+N$ are also of finite codimension, and

$$\codim(M+N)+\codim(M\cap N)=\codim M+\codim N.$$

The codimension of a submanifold $N$ of a differentiable manifold $M$ is the codimension of the tangent subspace $T_x(N)$ of the tangent space $T_x(M)$ at $x\in N$. If $M$ and $N$ are finite-dimensional, then

$$\codim N=\dim M-\dim N.$$

If $M$ and $N$ are differentiable manifolds, if $L$ is a submanifold of $N$ and if $f\colon M\to N$ is a differentiable mapping transversal to $L$, then

$$\codim f^{-1}(L)=\codim L.$$

The codimension of an algebraic subvariety (or an analytic subspace) $Y$ of an algebraic variety (analytic space) $X$ is the difference

$$\codim Y=\dim X-\dim Y.$$

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 $L$ of a vector space $V$ is equal to the dimension of any complement of $L$ in $V$, since all complements have the same dimension (as the orthogonal complement).

How to Cite This Entry:
Codimension. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Codimension&oldid=43514
This article was adapted from an original article by V.E. GovorovA.F. Kharshiladze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article