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In this proof of Jordan normal form in the Kaye and Wilson book, then for a transformation $T$ with minimal polynomial $m(x) = (x-e)^k$, they take a basis of $\texttt{ker}\;T$, extend it to a basis of $\texttt{ker}\;T^2$, ..., extend it to a basis of $\texttt{ker}\;T^k$. They then take the elements $a_1,...,a_n$ in $\texttt{ker}\;T^k$ but not $\texttt{ker}\;T^{k-1}$ then take $b_i = T(a_i)$ and claim the $b_i$,$a_j$ form a linearly independent set, so then they extend the list of the $b_i$ so that the span of the $a_j$, $b_i$ is $\texttt{ker}\;T^{k-1}$ not $\texttt{ker}\;T^{k-2}$. They then take $c_i = T(b_i)$ and carry on this process.

After the proof they say the crucial point is that the basis modification does give a basis. Then the lemma they prove is: If $ \{u_1,...,u_r \}$ is a basis for $\texttt{ker}\;T^j$ is extended to a basis of $\texttt{ker}\; T^{j+1}$ $\{u_1,...,u_r,v_1,...,v_s\}$ and to a basis $\{u_1,...,u_r,v_1,...,v_s,w_1,...,w_t\}$ of $\texttt{ker}\;T^{j+2}$ then $\{u_1,...,u_r,T(w_1),...,T(w_t)\}$ is a linearly independent subset of $\texttt{ker}\;T^{j+1}$.

I can't work out why they prove $\{u_1,...,u_r,T(w_1),...,T(w_t)\}$ is linearly independent - where does this come into the proof for Jordan normal form? For the proof for Jordan normal form surely they'd want to show that $\{w_1,...,w_t,T(w_1),...,T(w_t)\}$ is linearly independent since in the proof they claim the $b_i$,$a_j$ form a linearly independent set (in which case they wouldn't need to extend the basis of $u_i$ to a basis of $u_i$,$v_j$ and then to a basis $u_i$,$v_j$,$w_k$ - they wouldn't need the $u_i$ at all).

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It seems you changed the notion? They assume $m_T=x^k$...

I don't think they care about $a_j,b_i$ being linearly independent. What they want is, if their basis (of $V$) contains elements $a_1, \dots , a_{n_1}$ in $\ker T^l \setminus \ker T^{l-1}$ then also $g(a_1), \dots g(a_{n_1})$ is contained in the basis.

What they do is define their modified basis recursively starting at $\ker T^k$. At the $l$th stage they then apply $g$ to the elements of the basis contained $\ker T^{l+1} \setminus \ker T^{l}$ and extend the resulting elements to a basis of $T^{l}$ without messing around in the remaining basis elements of $T^{l-1}$(they need that this basis restricts also to a basis of $T^i$ for $i<l-1$) to be able to do the next step.

And I think the lemma should do this step.

What looks a bit strange to me is that they choose variables $a_i, \dots z_i$ for basis elements of the different powers of $T$ in the book.

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