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601-fis-p19 - The following exercise requires familiarity...

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Unformatted text preview: The following exercise requires familiarity with the definition of quotient space defined in Exercise 29 of Section 1.3 and Exercise 30 of Section 2.1. 22. Let T: V —. 2 be a linear transformation of a vector space V onto a vector space Z Define the mapping T: V/N(T) —> Z by T(v + N(T)) = T(v) for any coset v + MT) in V/N(T). (a) Prove that T is well-defined; that is, prove that if v + N(T) = v' + N(T), then T(v) = T(v’). (b) Prove that T is linear. (c) Prove that T is an isomorphism. (d) Prove that the diagram shown in Figure 2.3 commutes; that is, prove that T = Tn. T V ————— —>— z \\ 4 \ / / Figure 2.3 V/NiTl ‘9‘ sunny. u wuusva run-sunvu 2Q. Let T be a linear operator on a vector space V, and let W be a T-invariant ; subspace of V. Define T: V/W —+ V/W by T(v + W) = T(v) + W for any v + W in V/W. (a) Show that T is well-defined. That is, show that T(v + W) = T(v’ + W) whenever v + W = v' + W. (b) Prove that T is a linear operator on V/W. (e) Let n: V —> V/W be the linear transformation as defined in Exercise 30 of Section 2.1 by 17(0): 1) + W. Show that the diagram of Figure 5.5 is commutative, that is, "T = Tn. v———~——>v '7 72 T V/W ~————>-V/W Figure 5.5 In Exercises 27 through 29, T is a linear operator on a finite-dimensional vector space V and W is a nontrivial T-invariant subspace. 27. Let f: (t), g(t), and h(t) be the characteristic polynomials of T, Tw, and T, respectively. Prove that f(t)=g(t)h(t). Hint: Extend a basis y ={x1, x2,...,x,‘} for W to a basis B = {x,,x2,...,x,,} for V, show that a ={xH1 + W,...,x,, + W} is a basis for WW, and B, B. m" =(0 Ba)’ where B1 = [T]y and B3 = [7],. ...
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