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Vector space/Direct sum/Introduction/Section

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Let denote a field and let denote a -vector space. Let be a family of linear subspaces of . We say that is the direct sum of the , if the following conditions are fulfilled.

  1. Every vector has a representation

    where .

  2. for all .

If the sum of the is direct, then we write also instead of . For two linear subspaces

the second condition just means .


Let denote a finite-dimensional -vector space together with a basis . Let

be a partition of the index set. Let

be the linear subspaces generated by the subfamilies. Then

The extreme case yields the direct sum

with one-dimensional linear subspaces.


Let be a finite-dimensional -vector space and let be a linear subspace. Then there exists a linear subspace such that we have the direct sum decomposition

Let denote a basis of . We can extend this basis, according to fact, to a basis of . Then

fulfills all the properties of a direct sum.


In the preceding statement, the linear subspace is called a direct complement for (in ). In general, there are many different direct complements.