Let $V$ be a finite dimensional vector space and $A$ and $B$ two linear transformations on $V$ such that $A^{2}=B^{2}=0$ and $AB+BA=1$.
1) Prove that if $N_{A}$ and $N_{B}$ are respective null spaces of $A$ and $B$, then $N_{A}=AN_{B}$, $N_{B}=BN_{A}$, and $V=N_{A}\oplus N_{B}$.
2) Prove the dimension of $V$ is even.
3) Prove that if the dimension of $V$ is 2, then $V$ has a basis with respect to which $A$ and $B$ are represented by matrices $(0,1),(0,0)$ and $(0,0),(1,0)$ (sorry I do not know how to type matrices).
My main difficulty is to prove $N_{A}\oplus N_{B}=V$. It follows trivially that $N_{A}\cap N_{B}=\{0\}$, but I do not know how to prove $N_{A}\oplus N_{B}=V$, and I also do not know how to prove $n=2k$.