Vector space/Linear subspaces/Finite/Direct sum/Definition

Direct sum (linear subspaces)

Let ${}K$ denote a field, and let ${}V$ denote a ${}K$-vector space. Let ${}U_{1},\ldots ,U_{m}$ be a family of linear subspaces of ${}V$. We say that ${}V$ is the direct sum of the ${}U_{i}$ if the following conditions are fulfilled.

1. Every vector ${}v\in V$ has a representation

   ${}v=u_{1}+u_{2}+\cdots +u_{m}\,,$

   where ${}u_{i}\in U_{i}$.

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