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

  • #1
123
10
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.
 

Answers and Replies

  • #2
lavinia
Science Advisor
Gold Member
3,236
624
Try proving it first for the case ##n=1## so that the minimal polynomial is irreducible.
 

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