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 consider ${}\pi$ 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 ${}\operatorname {ker} {\left(\pi \times \kappa \right)}$.

Create a solution