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 the eigenvalues of the matrix A be A1, A2, . . A,n and the linearly w m w w w w w ma ndependent eigenvectors corresponding to these eigenvalues V1,V2,.. Vn. .... ww m ww (S [V1V2 . .. Vn]nxn) || by placing the vectors V1,V2,., Vn in each column of A is the eigenvalue of V1,V2,.., Vn placed on the w w m w m w w the S matrix; If the matrix diagonal, prove that, diagonal matrix. Also show that eAt mw wm m m w ww diag[A1 A2 . .. A,]nxn) with in the A = SAS-1 (A SeAt s-1 . %3D ww w w w w m
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).
A matrix A E M(3, R) that has as eigenvalues 1, -1, -1 associated respectively to the eigenvectors (50, 50, -50), (0,16,16) and (24,0, -24) corresponds to?
a. Show that for finite dimensional case the definition of point spectrum (infinite dimensional case) coincide with usual definition of eigenvalues set in finite dimensional case.
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
Define a linear operator I on P₂. by (Tp)(x) = xp'(x) + plo) Write a proof showing that find the eigenvalues and. eigenspaces of 7² 2 and say if I is diagonalizable
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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