Kernels and images of powers of operators Let $T$ be a linear operator on a vector space $V\rightarrow V$. Let $K_r$ and $W_r$ denote the kernel and image , respectively, of $T^r$.
(b) The following conditions might or might not hold for a particular value of $r$:
(1) $K_r=K_{r+1}$, (2)$W_r=W_{r+1}$, (3) $W_r\cap K_1=\{0\}$, (4) $W_1+K_r=V$.
Find all implications among the conditions (1)-(4) when V is finite dimensional.
I know 1, 2, and 3 are equivalent, but 4 is giving me a lot of trouble.
I know that if 1 holds, $K_r\cap W_r=\{0\}$, so $K_r\oplus W_r =V$, and since $W_r\subset W_1$, $K_r + W_1 =V$. I also know the 4 implies that there exists some subspace $W_1'$ such that $K_r\oplus W_1'=V$, but I don't know how to prove $W_1'=W_r$.
Note: there is a solution to this on this site here, but it uses notation that I don't understand.
 A: As long as you are allowed to assume that $V$ is finite dimensional, here is something that may help for (4):
Notice that for any fixed $r > 1$, 


*

*$\dim{W_{r}}+\dim{K_{r}} = \dim{V}$ (by rank-nullity)

*$\dim{W_{r+1}} \leq \dim{W_{r}}$, since $W_{r+1}\subseteq W_{r}$.

*$\dim{K_{r+1}} \geq \dim{K_{r}}$, since $K_{r} \subseteq K_{r+1}$.


Also, remember that that for any subspaces $A$,$B$, 
$$\dim{(A+B)} = \dim{A}+\dim{B} - \dim{(A \cap B)}.$$
Since you know that (1),(2), and (3) are equivalent, see if you can show that $\{(1),(2),(3)\} \Rightarrow (4)$, or that $(4) \Rightarrow \{(1),(2),(3)\}$.
If you assume (4), then every vector $\vec{v} \in V$ can be written as
a linear combination of a vector in $W_{1}$ and a vector in $K_{r}$. So if we take a vector $\vec{u} \in W_{r}$ then $\vec{u} = T^{r}(\vec{w} + \vec{k})$ where $\vec{w} \in W_{1}$ and $\vec{k} \in K_{r}$.
Thus
$$\vec{u} = T^{r}(\vec{w} + \vec{k}) = T^{r}(\vec{w}) + T^{r}(\vec{k}) = T^{r}(\vec{w}) + 0 = T^{r}(\vec{w}).$$
Now since $\vec{w} \in W_{1}$, $\vec{w} = T(\vec{v})$ for some $v \in V$, and so
$$T^{r}(\vec{w}) = T^{r}(T(\vec{v})) = T^{r+1}(\vec{v}) \in W_{r+1}.$$
This shows that $W_{r} \subseteq W_{r+1}$, and combined with point 2. above, shows that $W_{r}=W_{r+1}$ (and so $(4) \Rightarrow \{(1),(2),(3)\}$).
I'll leave the other direction for you. This proof really is a lot easier if you understand quotient spaces and can understand the other version; I'm not sure where you found this exercise, but you should make sure you work through all of the material leading up to it so you know you have all of the tools needed.
