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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Number 30
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
[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.
Consider the function T : P2 (R) → P2(R) given by T(a + bx + cx²) = (a – 6+ c) + bx + (-6 – 2c)a². Let T be a linear transformation and eigenvalues of T are A = 1 and A= 2. Find a basis for E2(T) and the geometric multiplicity of A = 2.
Let f and g be functions which are differentiable on R. For each of the following statements, determine if it is true or false. If it is true, then prove it. If it is false, then give a counterexample (you must prove that it is indeed a counterexample). (a) If f'(x) = g'(x) for all x, then f(0) = g(0). True False (b) If f'(x) > sin(x) + 2 for all x € R, then there is no solution to the equation ef(x) = 1. True False
Consider the linear transformation T: RR whose matrix A relative to the standard basis is given. 1-2 1 4 A = (a) Find the eigenvalues of A. (Enter your answers from smallest to largest.) (2₂₁, 2₂) = ([ (b) Find a basis for each of the corresponding eigenspaces. ={\ B₁ = B₂ = (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' =
3. Suppose L : R² → R² is a linear transformation. (a) If L(e,) = e, + €2 and L(e,) = e1 + €2, verify that e, + e, is an eigenvector of L. Find the corresponding eigenvalue. (b) Find the characteristic polynomial of L and then find another eigenvalue of L. (c) Find an eigenvector for the second eigenvalue you found in part (b).
4. Let E: P3 R2 be given by E (p(x)) = [P(2)] (a) Find a matrix that induces the transformation E. TH (b) Find a polynomial p(z) where E(p(x)) = [], or explain why this isn't possible. (c) Find a polynomial q(2) where E(q(z)) = [], or explain why this isn't possible. 8P (d) Find a nonzero polynomial r(z) where E(r(2)) = [] or explain why this isn't possible. (e) Describe the kernel of E. What is the dimension of the kernel? Find a basis for the kernel. (f) Describe the image of E. What is the dimension of the image? Find a basis for the image. (g) Is the transformation E one to one? Is it onto? Is it an isomorphism? Explain.
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.
For the matrix Z below, find the eigenvalues and eigenvectors. a. 1 0 -2 Z = 0 5 (-2 0 4 b. Verify if the function below is a linear transformation or not. T:P: →R defined by T (a + bx + cx³) = [2a – b - i. b+c] ii. L: R →R' defined by L = xy| Given the following transformation L(u)=Au, where u = (x, y) and the standard (or induced) с. matrix A is given as A i. Sketch the triangle connecting the points (2, -2), (2, -5) and (5, -2). Then sketch its transformation image if u represents the poit. ii. Identify the type of transformation shown above?

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