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

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In summary, the conversation is about a lemma in Schaum's outlines linear algebra that explains the decomposition of a linear operator into T-cyclic subspaces with corresponding T-annihilators. The writer finds the explanation frustrating and asks for help in understanding the proof. They offer to provide the proof from the book if necessary and thank anyone who can assist them.
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.

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

1. What is a lemma?

A lemma is a proven statement or proposition that is used as a stepping stone in the proof of a larger theorem or proposition. It is often a smaller, simpler statement that can be proved separately and then used to help prove a larger result.

2. What does it mean to decompose V to T-cyclic subspace?

Decomposing V to T-cyclic subspace means breaking down a vector space V into a direct sum of smaller subspaces, each of which is generated by a single vector under the action of a linear operator T. This is useful in understanding the structure of a vector space and its relationship to a linear operator.

3. How is a lemma on decomposition of V to T-cyclic subspace useful in mathematics?

A lemma on decomposition of V to T-cyclic subspace can be useful in many mathematical fields, such as linear algebra, functional analysis, and differential equations. It helps to simplify and clarify the structure of a vector space and the action of a linear operator, making it easier to prove more complex theorems.

4. What are some common techniques used to prove a lemma on decomposition of V to T-cyclic subspace?

There are various techniques that can be used to prove a lemma on decomposition of V to T-cyclic subspace, such as induction, contradiction, or direct proof. Depending on the specific lemma and its context, other techniques such as linear algebra or functional analysis tools may also be used.

5. Are there any real-world applications of a lemma on decomposition of V to T-cyclic subspace?

Yes, there are many real-world applications of a lemma on decomposition of V to T-cyclic subspace. For example, it can be used to study the behavior of dynamical systems, analyze data in signal processing and control theory, and understand the structure of physical systems in physics and engineering. It is also commonly used in the development of algorithms and computer programs for solving complex problems.

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