Take $V = K^{n}$. Let $\omega$ be a non-zero element of $\bigoplus_{k=1}^n \bigwedge^k V$, where we have excluded the summand $\bigwedge^0 V = K$. (1) Prove that there exists an $m > 1$ for which $\omega^m$, where the power is taken in the Grassmann algebra, is zero, and $\omega^{m-1}$ is not yet zero. (2) Find such an $\omega$ for which m of the preceding part is maximal. (Hint: pure non-zero tensors will have $m = 2$, which is certainly not maximal.)
I've figured out that Grassmann algebra is the same as exterior algebra, and that $m-1$ is probably the rank of $\omega$, and therefor in 2) we look for an $\omega$ of maximal rank. But I don't know how to prove it or how to find such a $\omega$.
Any help would be greatly appreciated.