Vector space/Basis/Subfamily/Projection/Example

Let ${}V$ be a finite-dimensional ${}K$-vector space and ${}v_{i}$, ${}i\in I$, a basis of ${}V$. For a subset ${}J\subseteq I$, set

${}V_{J}=\langle v_{i},\,i\in J\rangle \,,$

the linear subspace corresponding to ${}J$. Moreover, let

$p_{J}\colon V\longrightarrow V,\sum _{i\in I}a_{i}v_{i}\longmapsto \sum _{i\in J}a_{i}v_{i},$

the corresponding projection. The image of this projection is ${}V_{J}$. On ${}V_{J}$, this mapping is the identity, one can also consider this mapping as

$p_{J}\colon V\longrightarrow V_{J}.$

The kernel of this mapping is

${}\operatorname {kern} p_{J}=\langle v_{i},\,i\notin J\rangle \,.$