6) Suppose V is a finite-dimensionalwith dim V > 1 and T € L(V). Prove that{p(T) [p = F[x]} # L(V). This proof can be done without assuming the T has aneigenvalue. (If the field of scalars is R or Q, then not every operator need have aneigenvalue.)

Given : V is a finite dimensional vector space with dim V > 1 and T∈ LV To prove : p(T) : p∈Fx ≠…

Answered: Suppose V is a finite-dimensional with…

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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The answer has been given, please prove whether it is correct, Be sure to explain what you are doing in terms of the meaning of the words in the problems.
Consider T E L(V). (a) Suppose U is a subspace of V invariant under T. Prove that U is also invariant under p(T) for every polynomial p E P(IF). (b) Suppose v is an eigenvector of T with eigenvalue A. Suppose pE P(F). (b1) Show that p(T)v = p(1)v. (b2) Is v an eigenvector for p(T)? If so, for what eigenvalue?
Please solve anything you can solve
Let B be an n x n matrix over a field F, and suppose that B has minimal polynomial x2 + x + 1. (i) Show that B is invertible. (ii) (iii) (iv) Prove that the minimal polynomial of B-1 is also x? + x + 1. Prove that ifF = C and n is odd, then B is not similar to B-1. Prove that if F = R and n is even, then B is similar to B-1.
Given an n xn-matrix A over a field K, the following properties hold: (1) For every column vector b, there is a unique column vector z such that Ax = b iff the only solution to Ax = 0 is the trivial vector x = 0, iff det(A) # 0. (2) If det (A) #0, the unique solution of Ax=b is given by the expressions det (A¹,..., A-¹, b, Aj+¹, Ij: ...,A") = det (A¹,..., Ai-¹, Ai, Aj+¹,…, A¹) ¹ known as Cramer's rules. (3) The system of linear equations Ax = 0 has a nonzero solution iff det (A) = 0.
can you help me solve part b and c please?
Consider the following signals over the interval -1
Please solve question 4a and b
5
2. For each of the following linear operators T on a vector space V, test T for diagonalisability. If T is diagonal, find a basis 3 for V such that [T]8 is a diagonal matrix. (a) V = P2(R) and T(f(x)) = f'(x) + f"(x). (b) V = P2(R) and T(ax² + bx +c) = cx² + bx + a.
True or false
Let T: M₂x2 (R) → M2×2(R) be the linear operator given by the trans- pose, i.e. T(A) = A¹, (a) Compute det T, the determinant of T. (b) Compute pr(x), the characteristic polynomial of T. (c) Compute the eigenvalues of T.

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Suppose V is a finite-dimensional with dim V > 1 and T € L(V). Prove that {p(T)|p € F[x]} ‡ L(V). This proof can be done without assuming the T has an eigenvalue. (If the field of scalars is R or Q, then not every operator need have an eigenvalue.)