$f$ is a linear map on a $k$-vector space $M$ with $f^n=0$ for some $n>0$. $k$ is a field.
$k^d$ is $k$-algebra generated by $x_1,\ldots,x_d$ satisfy the relation $x_i x_j + x_j x_i =0$ for all $i,j=1, \ldots, d$.
Show that $\ker(f)$ is non-zero.
Thank you!