# Linear mapping/Mulled wine/Price and calories/Exercise

On the real vector space ${\displaystyle {}G=\mathbb {R} ^{4}}$
${\displaystyle \pi \colon G\longrightarrow \mathbb {R} ,{\begin{pmatrix}z\\n\\r\\s\end{pmatrix}}\longmapsto 8z+9n+5r+s,}$
${\displaystyle \kappa \colon G\longrightarrow \mathbb {R} ,{\begin{pmatrix}z\\n\\r\\s\end{pmatrix}}\longmapsto 2z+n+4r+8s.}$
We put ${\displaystyle {}\pi }$ as the price function and ${\displaystyle {}\kappa }$ as the caloric function. Determine a basis for ${\displaystyle {}\operatorname {ker} {\left(\pi \right)}}$, one for ${\displaystyle {}\operatorname {ker} {\left(\kappa \right)}}$ and one for ${\displaystyle {}\operatorname {ker} {\left((\pi \times \kappa )\right)}}$