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Question: let v be a vector space over a eld f...

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Let V be a vector space over a field F. Let T : V→V be a linear map. Let W be a subspace of V. We say W is T-invariant if for every w ∈ W, we have T(w) ∈ W. Let W be T-invariant. Then T restricts to a linear map W→W, which we denote by TW (given by TW(w) = T(w)). Let V/W be the quotient of V by W (linear aglebra)

3. Let V be a vector space over a field F. Let T: VVbe a linear map. Let W be a subspace of V. We say W is T-invariant if for every w є W, we have T(w) є W. Let w be T-invariant. Then T restricts to a linear map W → W, which we denote by Tw (given by Tw(w) = T(w)). Let V/W be the quotient of V by W (to recall its definition and some useful results see Exercise 31 on page 23, Exercise 35 on page 58, and Exercise 40 on page 79 of the textbook). (a) Show that T: v/w- V/W given by T(v+W) Tv) W is well-defined and linear (Being well-defined means that the definition makes sense. The reason we have to check this is because given v + w є VW, the formula for T(v + w) makes use of the representative v of the coset v + W. We need to make sure the output Tiv+ W) does not change if we choose a different representative for v+W. More explicitly, we need to make sure that if v + W-v, + w for some v, v, є V, then T(v) + W-T(v) + w.) 2 (b) Let V be finite-dimensional and W a nonzero proper T-invariant subspace. Denote the characteristic polynomials of T, Tw, and T respectively by f(t), g(t), and h(t). Show that f(t)-g(t)h(t). (Suggestion: Exercise 35 on page 58 can be useful.)

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