3. Suppose L : R² → R² is a linear transformation.(a) If L(ej) = e + €2 and L(e,)that e, + e, is an eigenvector of L. Find thecorresponding eigenvalue.= e1 + e2, verify(b) Find the characteristic polynomial of L andthen find another eigenvalue of L.(c) Find an eigenvector for the second eigenvalueyou found in part (b).

Answered: Image /qna-images/answer/2e95a4fb-a233-4539-ac67-6a1b000c032b.jpg

Answered: 3. Suppose L : R² → R² is a linear…

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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Find the linear transformation T: P) + P1) such that T has eigenvalues -2 and 3 associated with the polynomials p(x) = 3a + az and g(x) = -2b + bx, respectively. (Here a and b are arbitrary real numbers)
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Find the eigenvalues and eigenvectors of the linear transformation T: P₁ → P₁ given by T(ax + b) = (6a – 10b)x+ (2a − 3b). Eigenvalue: A = 2 Eigenvalue: A = Ex: 5 Eigenvector: 5x+ Eigenvector: x+1
The linear transformation described by the matrix A = T. 3] is a reflection across the line y = -x. Use this fact to find the two eigenvalues of Al and an eigenvector associated to each eigenvalue. You should be able to find the answers geometrically, without needing to do any calculations. Smaller eigenvalue = Associated eigenvector = Larger eigenvalue = Associated eigenvector = 0 Note: vectors are entered with "angle brackets", such as or .
Consider the linear transformation T: RR whose matrix A relative to the standard basis is given. 3 -1 A = 2 (a) Find the eigenvalues of A. (Enter your answers from smallest to largest.) (21, 22) = 4,5 (b) Find a basis for each of the corresponding eigenspaces. B₁ = {(1,-1) B2 (1,-2) = (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). A' =
[3x, + X2] 3. Let L: R? → R? be a linear transformation defined by L= 3 a. Find the matrix representation of L. b. Find the eigenvalues of L and the associated eigenvectors. c. Is A diagonalizable? Explain your answer.
4. Let A be a square matrix. (a) 1² is an eigenvalue of A? with the same eigenvector, v. Show that if A is an eigenvalue of A with corresponding eigenvector v, then (b) What is the characteristic polynomial of A2? Suppose that the characteristic polynomial of A is fa(1) = (A + 2)(A – 3). Suppose that v is an eigenvector of A². Is it necessarily true that v is also (c) an eigenvector of A? If so, then prove this fact; if not, then give an example that shows this is false.
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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
Consider the vector space R^4 and u = (u1, u2, u3, u4) ∈ R^4 . 1. Let Q : R^4 → R defined by Q(u) = |u3| + |u4| + sup{|u1|, |u2|}. Determine if Q is a norm, a seminorm or none of the two for R^4.
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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