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I am a grad student preparing for a quals. I am using H. and Kunze book.

the problem is:

let V be a n-dim vector space over F. and let [itex]B[/itex]={[itex]a_1,a_2,..., a_n[/itex]} be an ordered bases for V.

a- According to them 1, there is a unique Linear operator T on V such that

[itex]Ta_i=a_{(i+1)}[/itex] , i=1,....,n. and [itex]Ta_n=0[/itex].

what is the matrix [itex]A[/itex] of [itex]T[/itex] in the ordered bases B.

b- prove that [itex]T^n=0, and\ \ \ T^{n-1}≠ 0[/itex].

c- Let S be any linear operator on V such that [itex]S^n=0\ \ \ but\ \ \ S^{n-1}≠0[/itex]. Prove that there is an ordered bases [itex]B'[/itex] such that the matrix of [itex]S[/itex] in the bases [itex]B'[/itex] is the matrix [itex]A[/itex] of part (a).

Solution Attempt.

Obviously we have for (a)

[itex]A= \begin{bmatrix}

0 & 0&... & 0 &0\\

1 & 0 & 0&....&0 \\

0&1&0&.....&0\\

.\\

.\\

0&0&....&1&0

\end{bmatrix}[/itex]

for (b) its obvious.

my problem is with the last question. I tried to justfy it by two ways, the first one is to find an ivertible linear transformation [itex]U:V→V[/itex] such that [itex]S=UTU^{-1}[/itex], then we will be done and such a bases exists. The second way is that I am trying to show the follwing:

there exist at least on vector in the bases [itex]B[/itex] such that [itex]S^ia_i≠0[/itex] for [itex]i=1,..,n-1.[/itex] and I am considering the set [itex]B'=[/itex]{[itex]a_i, Sa_i,S^2a_i,....,S^{n-1}a_i[/itex]}. Note if we proved [itex]B'[/itex] is a bases, then [itex]_{B'}=A[/itex].i.e. the matrix of [itex]S[/itex] relative to the bases [itex]B'[/itex] is A.

Unfotunatly, I could not get to an end with both ways.

Am I doing the right thing? Any suggestions?.

Thank you in Advance.

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# A linear Trans. prob.

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