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Here's a homework problem I never got around to doing when I took the course. I'll state it in full and share what what I know and where I'm confused (more in the latter category than the former I'm afraid):

Let $S:W\rightarrow W$ be a linear transformation, and let $A_k=\text{im}(S^k)$ and $B_k=\text{ker}(S^k)$.

a. Show that $A_{k+1}\subseteq A_k$ and that $A=\bigcap A_i$ is a subspace, and similarly that $B_k\subseteq B_{k+1}$ and that $B=\bigcup B_i$ is a subspace.

b. Show that for $\text{dim}V\lt \infty$, $A=A_k$ for some k, and $B=B_k$ for some possibly different $k$, and find a bound on these independent of $S$. Further show that $V=A\oplus B$.

c. Show that $T$ maps $A$ to $A$ and $B$ to $B$, and further that if $\text{dim}V\lt \infty$ that $S|_A$ is nilpotent, and $S|_C$ is an isomorphism. Then I want to use this to show that any matrix in $\mathbb{F}^{n\times n}$ is similar to a matrix of the form $$\begin{bmatrix} Q &0 \\ 0 & R \end{bmatrix}$$, with $Q$ invertible and $R$ nilpotent.

So I'm good on a, thankfully! Intuitively I believe I understand the first part b, but could use some help making the argument more formal. It seems as though the bound should be the dimension of $V$, and in some sense I "feel" this is true since repeatedly applying $S$ had better kill some stuff then stop killing it, or kill everything eventually. Certainly it can't preserve anything and then kill it later on. So the slowest everything can die would be dimension of $V$ steps I think. And $A$ and $B$ are related by rank-nullity. But again, I'd love to see this part done formally without all my lexical hand-waving.

The rest I'm pretty much at a loss. I'm not going to demand full solutions to something I don't have a clue on, but hints and patient guidance toward filling in these pieces would be great. It certainly makes sense that $A$ and $B$ should be complementary. But showing how to decompose a $v$\inV such that it's the sum of an element in each constituent is escaping me. Certainly, there are bases of adequate dimension in some $S^n$, but I'd want them to have trivial intersection, and I'm failing at this...

Any and all help I'd be forever grateful for. Or tips on how I can make the question more precise so I don't leave potential answerers wandering in the dark.

I'm going to tag this as homework, but I do stress that it's from a course from a previous semester.

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Refer Carl D Meyer "Matrix Analysis" pgnos: 394 to 403. He explains every thing you have posed in very good details. The book is available online. –  dineshdileep Oct 22 '12 at 6:09
    
Instead of hand-waving you should just start to do something. Every part of this exercise is straight-forward. –  Martin Brandenburg Oct 22 '12 at 8:03
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@MartinBrandenburg The handwaving comes from failing at the doing. I wouldn't take the time to post the question if I could just do it. Sometimes the nike slogan just isn't enough to get me through. I'm very happy I now know it's straightforward. The next goal I guess is knowing how to solve the problem. –  AsinglePANCAKE Oct 22 '12 at 8:41
    
@dineshdileep: Thank you! That resource was quite helpful. –  AsinglePANCAKE Oct 22 '12 at 8:42

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