9. Let T be an operator on an n-dimensional vector space V over a field F.(a) If T has n distinct eigenvalues prove that T is diagonalizable.(b) If T is diagonaziable prove that the characteristic polynomial of T splits over F.(c) Give an example of an operator T such that the characteristic polynomial of Tsplits but T is not diagonalizable.

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Answered: 9. Let T be an operator on an…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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(b) Let f V -» V be any linear map of vector spaces over a field K. Recall that, for any polynomial p(X) = 0 X e K[X] and any v E V, p(X) vp(v) = ef*(v). i-0 The kernel of p(X) is defined to be Ker(p(X)) {v e V : p(X) v 0}. (b.1) Show that Ker(p(X)) is a linear subspace of V. When p(X) = X - A where E K, explain that Ker(P(X)) is the eigenspace of f with respect to A. (b.2) Let p(X) and q(X) be polynomials in K[X] so that gcd(p(X), q(X)) = 1. Show that Ker(p(X)q(X)) — Ker(p(X)) + Кer(4(х)) is a direct sum of subspaces. (Hint: Use the fact that if gcd(p(X), q(X)) = 1 then there exist a(X), Ь(X) € К|X] so that a(X)p(X) + b(X)q(X) %3D 1.) (b.3) Let c(X) e K[X] be any nonzero polynomial so that c(f) = 0 (for example c(X) is the characteristic polynomial of f). Suppose с (X) — р1 (X)р:(X)... Pm(X) where each p,(X) e K[X] and gcd(pg(X), p3(X))= 1 for all pairs 1 i
at most 2. Find a basis {p(x), q(x)} for the vector space {f(x) = P₂[x] | f'(-8) = f(1)} where P₂[x] is the vector space of polynomials in x with degree You can enter polynomials using notation e.g., 5+3xx for 5 + 3x². p(x) = , 9(x) =
Let V be a vector space with dim(V) = 3. Explain why any T: V → V has at least one real eigenvalue.
Let P2(R) be the vector space of polynomials over R up to degree 2. ConsiderB ={1 + x − 2x^2 , −1 + x + x^2, 1 − x + x^2},B' ={1 − 3x + x^2, 1 − 3x − 2x^2, 1 − 2x + 3x^2}.(a) Show that B and B' are bases of P2(R).(b) Find the coordinate matrices of p(x) = 9x^2 + 4x − 2 relative to the bases B and B'.(c) Find the transition matrix P_B=→B'(d) Verify that, [x(p)]B'= P_B→B' [x(p)B].
Let K = Z5, the field of integers modulo 5. (You can read about fields in Chapter 1.8 of the textbook). Consider the vector space P2 of polynomials of degree at most 2 with coefficients in K. Are the polynomials x + x + 3, 4x² + 2x + 4, and 4x + 1 linearly independent over Z5? linearly independent v If they are linearly dependent, enter a non-trivial solution to the equation below. If they are linearly independent, enter any solution to the equation below. (х2 + х + 3)+ о (4x2 + 2x + 4)+ 0 (4х + 1) — 0.
Let V be a finite-dimensional vector space over F with T in L(V,V) invertible and lambda in F \ {0}. Prove that lambda is an eigenvalue for T if and only if lambda-1 is an eigenvalue for T-1.
V n-dimensional vector space T: V → V Let it be the linear transformation. Let the eigenvalue of this linear transformation be λ and the eigenvector corresponding to this eigenvalue x ≠ 0 So which of the following is true? a)There is only one λ value corresponding to the vector X. b)There is an infinite eigenvector for λ eigenvalue c)linear transform has real eigenvalue as much as the degree of the characteristic polynomial d)The matrix representation of the linner transformation is the invertible matrix A and the eigenvalue of A-1 is λ-1 ,with an eigenvalue λ.
Let A be an n×n matrix whose characteristic polynomial splits, γ be a cycle of generalized eigenvectors corresponding to an eigenvalue λ, and W be the subspace spanned by γ. Define γ’ to be the ordered set obtained from γ by reversing the order of the vectors in γ. (a) Prove that [TW]γ’ = ([TW]γ)t. (b)Let J be the Jordan canonical form of A. Use (a) to prove that J and Jt are similar. (c) Use (b) to prove that A and At are similar.
Let P3(R) be the vector space of all real polynomials of at most degrees 3 with the inner product defined as (f,g) = [ * f(x)g(x)dx. (a) Find an orthonormal basis B for P3 (b) Consider the linear operator T : P3 → P3 defined as T(p) = p". Find the charac- teristic polynomial and the minimal polynomial for T. (c) Find the matrix of T from part (b) with respect to B from part (a).
Consider the vector space V of all the real polynomials of degree less than or equal to three, p(x) = ax^3 + b x^2 + c x + d, and consider the linear transformation T: V ------> V given by the derivative, i.e., T( p ) = p' = 3ax^2 + 2 b x + c. Find the eigenvalues and eigenvectors of this transformation.
3. Let T be a linear operator on the finite – dimensional vector space V over the field F. Suppose that the minimal polynomial for T decomposes over F into a product of linear polynomials. Then there is a diagonalizable operator D on V and a nilpotent operator N on V such that (1) T (ii) DN D + N, ND. The diagonalizable operator D and the nilpotent operator N are uniquely determined by (i) and (ii) and each of them is a polynomial in T.

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