Fruit salad/Minerals and vitamins/Example

A healthy breakfast starts with a fruit salad. The following table shows how much vitamin C, calcium, and magnesium various fruits have (in milligrams with respect to 100 grams of the fruit).

	apple	orange	grapes	banana
vitamin C	${}12$	${}53$	${}4$	${}9$
calcium	${}7$	${}40$	${}12$	${}5$
magnesium	${}6$	${}10$	${}8$	${}27$

This table yields a mapping, which assigns to a ${}4$-tuple ${}{\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\\x_{4}\end{pmatrix}}$, representing the used fruits, the content of the resulting fruit salad with respect to vitamin C, calcium, and magnesium, in the form of a ${}3$-tuple ${}{\begin{pmatrix}y_{1}\\y_{2}\\y_{3}\end{pmatrix}}$. This mapping can be described with the matrix

${\begin{pmatrix}12&53&4&9\\7&40&12&5\\6&10&8&27\end{pmatrix}}$

using matrix multiplication as

${\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\\x_{4}\end{pmatrix}}\longmapsto {\begin{pmatrix}12&53&4&9\\7&40&12&5\\6&10&8&27\end{pmatrix}}{\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\\x_{4}\end{pmatrix}}={\begin{pmatrix}12x_{1}+53x_{2}+4x_{3}+9x_{4}\\7x_{1}+40x_{2}+12x_{3}+5x_{4}\\6x_{1}+10x_{2}+8x_{3}+27x_{4}\end{pmatrix}}={\begin{pmatrix}y_{1}\\y_{2}\\y_{3}\end{pmatrix}}.$