T be a linear operator (I think they mean "(adsbygoogle = window.adsbygoogle || []).push({}); LetT be a linear...") on a finite-dimensional vector space V, and let [itex]W_1[/itex] be a T-invariant subspace of V. Let [itex]x \in V[/itex] such that [itex]x \notin W_1[/itex]. Prove the following results:

- There exists a unique monic polynomial [itex]g_1(t)[/itex] of least positive degree such that [itex]g_1(T)(x) \in W_1[/itex].
- If h(t) is a polynomial for which [itex]h(T)(x) \in W_1[/itex], then [itex]g_1(t)[/itex] divides h(t).
- [itex]g_1(t)[/itex] divides the minimal and characteristic polynomials of T.
- Let [itex]W_2[/itex] be a T-invariant subspace of V such that [itex]W_2 \subseteq W_1[/itex], and [itex]g_2(t)[/itex] be the unique monic polynomial of least degree such that [itex]g_2(T)(x) \in W_2[/itex]. Then [itex]g_2(t)[/itex] divides [itex] g_2(t)[/itex].

- If [itex]\beta[/itex] is a basis of V, and [itex]\beta _{W_1}[/itex] is a basis of [itex]W_1[/itex] such that [itex]\beta _{W_1} \subseteq \beta[/itex], then define [itex]W_1 \prime = Span(\beta - \beta _{W_1})[/itex]. [itex]V = W_1 \oplus W_1 \prime[/itex]. [itex]\forall v \in V, \exists w_1 \in W_1, w_1 \prime \in W_1 \prime[/itex] such that [itex]v = w_1 + w_1 \prime[/itex].

[tex]g_1(T)(x) = g_1(T)(w_1 + w_1 \prime) = g_1(T)(w_1) + g_1(T)(w_1 \prime)[/tex]

Now,ifthe restriction of T to [itex]W_1 \prime[/itex] were an operator on [itex]W_1 \prime[/itex], then there would be a unique monic polynomial of least degree such that [itex]g_1(T)(w_1 \prime) = 0[/itex], namely the minimal polynomial of T restricted to [itex]W_1 \prime[/itex]. Then,if[itex]g_1(t)[/itex] is a polynomial over the same field that underlies [itex]W_1[/itex], I can assert that [itex]g_1(T)(w_1) \in W_1[/itex], and thus the result is proved. Can I prove these "if"s? If not, is there another way to prove the result? I haven't looked at the rest of it yet.

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# Homework Help: Stuck on this Algebra Problem

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