K^n/Linear form/Example

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

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

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

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

are the easiest linear forms.

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