K^n/Linear form/Price/Example

In a shop, there are ${\displaystyle {}n}$ different products to buy, and the price of the ${\displaystyle {}i}$-th product (with respect to a certain unit) is ${\displaystyle {}a_{i}}$. A purchase is described by the ${\displaystyle {}n}$-tuple

${\displaystyle \left(x_{1},\,x_{2},\,\ldots ,\,x_{n}\right),}$

where ${\displaystyle {}x_{i}}$ is the amount of the ${\displaystyle {}i}$-th product bought. The price for the purchase is hence ${\displaystyle {}\sum _{i=1}^{n}a_{i}x_{i}}$. The price mapping

${\displaystyle \mathbb {R} ^{n}\longrightarrow \mathbb {R} ,\left(x_{1},\,x_{2},\,\ldots ,\,x_{n}\right)\longmapsto \sum _{i=1}^{n}a_{i}x_{i},}$

is linear. This means, for example, that if we do first the purchase ${\displaystyle {}\left(x_{1},\,x_{2},\,\ldots ,\,x_{n}\right)}$ and then, a week later, the purchase ${\displaystyle {}\left(y_{1},\,y_{2},\,\ldots ,\,y_{n}\right)}$, then the price of the two purchases together is the same as the price of the purchase ${\displaystyle {}\left(x_{1}+y_{1},\,x_{2}+y_{2},\,\ldots ,\,x_{n}+y_{n}\right)}$.