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The codimension describes the complement to the dimension in various areas of mathematics . So in -dimensional space the sum of dimension and codimension of an object is the same.In three-dimensional space, a surface (dimension: 2) has codimension 1, a line (dimension: 1) codimension 2 and a point (dimension: 0) has Codimension 3.


If a vector space is over any field and is a subspace of , then the codimension of in becomes through

thus defined as the dimension of the factor space .


  • It always applies
Is finite dimensional, so it is
  • Is a complementary space of in , i.e. H. , so is
  • If there are two subspaces, then it always holds
  • If there are subspaces, then


A plane has dimension 2. In a three-dimensional space it has codimension 1 and in a four-dimensional space it has codimension 2. A point has codimension 1 in a straight line and codimension 2 in a plane. A hyperplane always has codimension 1, the dimension of the hyperplane is always 1 smaller than the dimension of the surrounding space.