# I Proving a lemma on decomposition of V to T-cyclic subspace

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1. Mar 16, 2017

I am reading Schaum's outlines linear algebra, and have reached an explanation of the following lemma:
Let $T:V→V$ be a linear operator whose minimal polynomial is $f(t)^n$ where $f(t)$ is a monic irreducible polynomial. Then V is the direct sum
$V=Z(v_1,T)⊕...⊕Z(v_r,T)$
of T-cyclic subspaces $Z(v_i,T)$ with corresponding T-annihilators
$f(t)^{n_1}, f(t)^{n_2},..., f(t)^{n_r}, n=n_1≥n_2≥...≥n_r$
Any other decomposition of V into T-cyclic subspaces has the same number of components and the same set of T-annihilators.

Now, it seems that while writing the explanation for this lemma the writer forgot the concept of explaining one's arguments when presenting a proof, which resulted in a long explanation which goes from one conclusion to the next without explaining how, which naturally was rather frustrating. If anyone could present the proof for this lemma to me, I would be very grateful.
Should you require it I can also copy the proof from the book (page 343 problem 10.31).
Thanks in advance to all the helpers.

2. Mar 21, 2017

### PF_Help_Bot

Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.

3. Mar 24, 2017

### lavinia

Try proving it first for the case $n=1$ so that the minimal polynomial is irreducible.