Let TE L(F2,1) such thatT(for any z, w E F.(a) Find all the eigenvalues of T and their corresponding eigenspaces.(b) Find an eigenbasis for F2.1 and use that basis to get the matrix for T withrespect to the eigenbasis.(c) List 4 different subspaces of F2- which are invariant under T. Explain whythey are invariant.

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Answered: Let T E L(F<") such that T( for any z,…

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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(3) Let matrix A have eigenvectors . . associated with eigenvalues Xo = 4, A₁ = 2, A₂ = 0), respectively. Suppose some vector 7 is: 7 = №n +27₁ +30¹₂ Compute Ar. Give your result as a linear combination of eigenvectors ₁.
Let T be a linear operator defined on P3 by T(a0+a1x+a2x^2)=a0+a1(x-1)+a2(x-1)^2. The third row of the matrix associated with T with respect to base 1, x, x^2 is: a) (0,0,1) b) other response c) (1,-2,1) d) (1,0,0) e) (-1,1,0)
Let A be an n x n matrix and let λ be an eigenvalue of A. The collection of all eigenvectors corresponding to A, together with the zero vector, is called the eigenspace of A and is denoted by EA. qs)Why is it important that this definition includes the phrase "together with the zero vector"? (i.e., what would go wrong if this phrase was not included?)
Please use the following: - The union of the bases for all eigenspaces is a maximal linearly independent set of eigenvectors. - The matrix (or linear operator) is diagonalizable if and only if the number of vectors in this set is equal to the size of the matrix (or the dimension of the vector space).
If wis an eigenvector of A, how does the vector Au compare geometrically to the vector ? O A is a reflection of in the z-axis O A is a reflection of win the y-axis O Aw is parallel to w but may have a different length O Aw is a rotation of
7. Let A be an n x n Hermitian matrix. (a) Show that (Au, v) = (u, Av) for any n-dim vectors u and v. Here (, ·) denotes the inner product. (Hint: rewrite the inner product as matrix multiplication.) (b) Let x be an eigenvector associated to the eigenvalue A. Based on part (a), show that A(x, x) = X(x, x). (c) As in part (b), show that = X; that is, the eigenvalue A is real. 1
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.

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