Definition: Direct Sum of Vector Spaces

Let \(F\) be a field and let \(V\) be a vector space over \(F\). Further, let \(U_{1},\ldots ,U_{m}\) be a family of subspaces of \(V\). We say that \(V\) is a direct sum of the \(U_{i}\), if the following two properties are fulfilled:

  1. \(U_{i}\cap \left(\sum _{j\neq i}U_{j}\right)=0\) for all \(i\).
  2. \(U_{i}\cap \left(\sum _{j\neq i}U_{j}\right)=0\) for all \(i\).

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Adapted from CC BY-SA 3.0 Sources:

  1. Brenner, Prof. Dr. rer. nat., Holger: Various courses at the University of Osnabrück