Linear mapping/Mulled wine/Price and calories/Exercise

On the real vector space ${}G=\mathbb {R} ^{4}$

of mulled wines we consider the two linear maps

$\pi \colon G\longrightarrow \mathbb {R} ,{\begin{pmatrix}z\\n\\r\\s\end{pmatrix}}\longmapsto 8z+9n+5r+s,$

and

$\kappa \colon G\longrightarrow \mathbb {R} ,{\begin{pmatrix}z\\n\\r\\s\end{pmatrix}}\longmapsto 2z+n+4r+8s.$

We put ${}\pi$ as the price function and ${}\kappa$ as the caloric function. Determine a basis for ${}\operatorname {ker} {\left(\pi \right)}$, one for ${}\operatorname {ker} {\left(\kappa \right)}$ and one for ${}\operatorname {ker} {\left((\pi \times \kappa )\right)}$

(Do not mind that there may exist negative numbers. In a mulled wine of course the ingredients do not come in with a negative coefficient. But if you would like to consider for example, in how many ways you can change a particular recipe, without changing the total price or the total amount of energy, then the negative entries make sense).

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