K^n/Linear form/Price/Example

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

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

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

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

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