<s>[INST] How to list all possible dimension of $\ker{T},\ker{T^2},...,\ker{T^{k-1}}$ and the corresponding canonical forms? Let $V$ be $5$-dimension vector space, and $T:\ V\rightarrow V$ a nilpotent linear transformation of order (index) $k$ where $1\le k\le 5$. How to list all possible dimension of $\ker{T},\ker{T^2},...,\ker{T^{k-1}}$ and the corresponding canonical forms? 
In my personal opinion, I feel like all dims are possible. For example $T=I$, then $\ker{T}=\ker{T^2}=...=\ker{T^{k-1}}=\{0\}$. So dimension is $1$. But I have no idea for the canonical form.
As for other dimensions, since it's in increasing order, if $\dim(\ker{T^k})=n$, then $\dim(\ker{T^{k-1}})\le n$. 
The problem is I have no firm proof of this and I don't know how to provide general canonical form. Could someone give any insight?
 [/INST] You really should specify what you mean by "canonical form". I will assume that you meant rational canonical form since you did not specify that you were dealing with a complex vector space. In this particular problem, because of the nilpotency, these are also the Jordan forms, but that is just luck, the distinction of which canonical form you are talking about is important generally. 
Also, I will leave the computation of the dimensions of the kernels to you. With the canonical forms written down, it will be straightforward.
$k=1:$ $T$ must be the zero transformation. The zero matrix is its canonical form.
$k=2:$ The minimal polynomial is $x^2$. The invariant factors are then $x,x,x,x^2$ or $x,x^2,x^2$. The corresponding rational canonical forms are respectively, $$\begin{pmatrix}
0 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 \\
\end{pmatrix} \textrm{ or } \begin{pmatrix}
0 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 \\
\end{pmatrix}_.$$
$k=3:$ The minimal polynomial is $x^3$. The invariant factors are then $x,x,x^3$ or $x^2,x^3$. The corresponding rational canonical forms are respectively, $$ \begin{pmatrix}
0 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 \\
\end{pmatrix} \textrm{ or } \begin{pmatrix}
0 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 \\
\end{pmatrix}_.$$
$k=4:$ The minimal polynomial is $x^4$. The invariant factors are $x,x^4$. The corresponding rational canonical form is, $$ \begin{pmatrix}
0 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 \\
\end{pmatrix}_.$$
$k=5:$ The minimal polynomial is $x^5$. The invariant factor is $x^5$. The corresponding rational canonical form is, $$ \begin{pmatrix}
0 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 \\
\end{pmatrix}_.$$
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