K^n/Linear form/Example

A linear form on ${}K^{n}$ is of the form

$K^{n}\longrightarrow K,\left(x_{1},\,\ldots ,\,x_{n}\right)\longmapsto \sum _{i=1}^{n}a_{i}x_{i},$

for a tuple ${}\left(a_{1},\,\ldots ,\,a_{n}\right)$. The projections

$p_{j}\colon K^{n}\longrightarrow K,\left(x_{1},\,\ldots ,\,x_{n}\right)\longmapsto x_{j},$

are the easiest linear forms.

The zero mapping to ${}K$ is also a linear form, called the zero form.