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

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.

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#### PF_Help_Bot

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#### lavinia

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