5. Let V be a vector space over C and let T : V →V be a linear transformation.(a) Give the definition of an eigenvector of T and the definition of an eigenvalue of T.2(b) If u, w E V are eigenvectors for V with eigenvalues d and µ respectively such that A + µ,prove that u+ w is not an eigenvector of T.(c) For all y E C, prove thatVy = {u € V : T(u) = yu}is a subspace of V.(d) If y E C and Vy = {0}, is y an eigenvalue of T? Justify your answer.(e) If the characteristic polynomial of T isPr(t) = (t – 1)(t – 3)(t+ i)(t – i)find the minimum polynomial of T. Explain your answer carefully.

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Answered: 5. Let V be a vector space over C and…

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Let Pn denote the vector space of polynomials in the variable x of degree n or less with real coefficients. Let D : P3 - P2 be the function that sends a polynomial to its derivative. That is, D(p(x)) = p' (x) for all polynomials p(x) = P3. Is D a linear transformation? Let p(x) = a³x³ + ²x² + a₁ + a₁ and q(x) = b3x³ + b₂x² + b₁x + b₁ be any two polynomials in P3 and c E R. a. D(p(x) + q(x)) = . (Enter a3 as a3, etc.) D(p(x)) + D(q(x)) = Does D(p(x) + q(x)) = D(p(x)) + D(q(x)) for all p(x), q(x) = P3? choose b. D(cp(x)) = + c(D(p(x))) = = Does D(cp(x)) = c(D(p(x))) for all CER and all p(x) = P3? choose ✪ c. Is D a linear transformation? choose ◆
(3) Let V be a finite-dimensional vector space and TE L(V). Let A1,...,Am denote the distinct nonzero eigenvalues of T. Prove that T is diagonalizable if and only if dim E(A1,T) + ...+ dim E(m, T) = dim range T.
Let V be a vector space with dim(V) = 3. Explain why any T: V → V has at least one real eigenvalue.
Let B = {1,x, r²} be the standard ordered basis for P2(R). Let T be the linear operator on P2(IR) defined by T(f(x)) = 2f'(x) + f(1)x. %3D Find the eigenvalues and eigenvectors of T.
Consider the linear transformation T: R → Rn whose matrix A relative to the standard basis is given. 2 A-[4] = 7 (a) Find the eigenvalues of A. (Enter your answers from smallest to largest.) -([ (2₁, 2₂) = (b) Find a basis for each of the corresponding eigenspaces. {[ B₁ B2 = A' = = (c) Find the matrix A' for T relative to the basis B', where B' is made up of the basis vectors found in part (b). 1
Let A be the matrix of the linear transformation T:R2→R2 that reflects each point across the x1-axis. What are the eigenvectors and corresponding eigenvalues of A?
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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.
Consider the linear transformation T: R^ →R" whose matrix A relative to the standard basis is given. [³ A = (a) Find the eigenvalues of A. (Enter your answers from smallest to largest.) (1₁, 12) = 4,5 B₂ 3-2 (b) Find a basis for each of the corresponding eigenspaces. B₁ = 1,2 128 1,1 = A' = 1 6 (c) Find the matrix A' for T relative to the basis B', where B' is made up of the basis vectors found in part (b). X 5 5 1 2 ↓ 1
Let T: R³ R³ denote the linear transformation defined by T() = proj, and let S: R³ R³ denote the linear transformation defined by → S(*) = 7 proj The standard matrix Ar of T has eigenvalues and (ordered from smallest to largest). The eigenvalue Questions about T has basic eigenvector(s), and the eigenvalue has basic eigenvector(s). The standard matrix Ardiagonalizable. has Questions about S The standard matrix As of S has eigenvalues and (ordered from smallest to largest). The eigenvaluer basic eigenvector(s), and the eigenvalue has basic eigenvector(s) The standard matrix As diagonalizable. Questions about So T The only eigenvalue of As-r of Sis It has basic eigenvector(s) The standard matrix As diagonalizable.
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