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Let T E End (V) and let W≤ V be a T-invariant subspace. Denote by a. Prove that I is well-defined and linear. T: V W V W : x → T(x) b. 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). = c. Find g(t) and h(t) if T: R³ → R³ is such that its matrix in the standard basis is the matrix Ar below, and W = We₂ 2 AT --(13) -7

Let T E End (V) and let W≤ V be a T-invariant subspace. Denote by a. Prove that I is well-defined and linear. T: V W V W : x → T(x) b. 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). = c. Find g(t) and h(t) if T: R³ → R³ is such that its matrix in the standard basis is the matrix Ar below, and W = We₂ 2 AT --(13) -7

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Let T E End (V) and let W≤ V be a T-invariant subspace. Denote by a. Prove that I is well-defined and linear. T: V W V W : x → T(x) b. 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). = c. Find g(t) and h(t) if T: R³ → R³ is such that its matrix in the standard basis is the matrix Ar below, and W = We₂ 2 AT --(13) -7
Transcribed Image Text:Let T E End (V) and let W≤ V be a T-invariant subspace. Denote by a. Prove that I is well-defined and linear. T: V W V W : x → T(x) b. 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). = c. Find g(t) and h(t) if T: R³ → R³ is such that its matrix in the standard basis is the matrix Ar below, and W = We₂ 2 AT --(13) -7
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