Vector space/Direct sum/Introduction/Section
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.
- Every vector
has a representation
where .
- for all .
If the sum of the is direct, then we also write 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
In the preceding statement, the linear subspace is called a direct complement for
(in ).
In general, there are many different direct complements.