K^n/ith projection/Linear mapping/Example

Let ${\displaystyle {}K}$ denote a field, and let ${\displaystyle {}K^{n}}$ be the ${\displaystyle {}n}$-dimensional standard space. Then the ${\displaystyle {}i}$-th projection, this is the mapping

${\displaystyle K^{n}\longrightarrow K,\left(x_{1},\,\ldots ,\,x_{i-1},\,x_{i},\,x_{i+1},\,\ldots ,\,x_{n}\right)\longmapsto x_{i},}$

is a ${\displaystyle {}K}$-linear mapping. This follows immediately from componentwise addition and scalar multiplication on the standard space. The ${\displaystyle {}i}$-th projection is also called the ${\displaystyle {}i}$-th coordinate function.