Vector space/Basis/Index partition/Direct sum/Example

Let ${}V$ denote a finite-dimensional ${}K$-vector space together with a basis ${}v_{1},\ldots ,v_{n}$. Let

${}\{1,\ldots ,n\}=I_{1}\uplus \ldots \uplus I_{k}\,$

be a partition of the index set. Let

${}U_{j}=\langle v_{i},\,i\in I_{j}\rangle \,$

be the linear subspaces generated by the subfamilies. Then

${}V=U_{1}\oplus \cdots \oplus U_{k}\,.$

The extreme case ${}I_{j}=\{j\}$ yields the direct sum

${}V=Kv_{1}\oplus \cdots \oplus Kv_{n}\,$

with one-dimensional linear subspaces.