I hope you can help me, I'm very new to linear algebra, I am given the linear transformation $T:V\rightarrow V$ that fullfills $T^2 = 0$, what can be said of the relationship that exists between Im$(V)$ and $\ker(V)$? What I did so far is the following:
$T(Tx) = 0 \implies Tx \in \ker(T) \wedge Tx \in $Im$(T) \implies Tx \in \ker(T)\bigcap $ Im$(T)$
but $Tx$ should be equal to $0$ according to transformation's definition because $Tx \in $ Im$(T)$ also belongs to $V$ and given the transformation's definition $T(Tx) = 0$ and picking $w=Tx$ whe could say $T(w) = 0$ $ \forall$ $w \in V$ so, can we conclude that $\ker(T)\bigcap$ Im$(T) = 0?$, so $\ker(T)$ and Im$(T)$ are disjoint sets? can we conclude anything else? am I right?
thanks for any replies, cheers.