K^n/ith projection/Linear mapping/Example

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

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

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