5. (*) The linear transformation L: V → V has eigenvalues A1 and A2 such that A1 2. Recall for i = 1, 2 VA: := {v € V : L(v) = A;v} and define for i = 1,2 Vdg := {v € V : there is u e V, such that L(v) = d;v + u}. (This is called a generalised eigenspace – think of the vectors as "almost" eigenvectors, or eigenvectors up to an "error" term.) a) Prove that VA1A1 is a subspace of V and that VA, C VA1,d b) Prove that V1a O Va2 = {0}. 2 1 c) For A = 2 E Mat3x3(R), V = R³ and L = LA, find A1, 2 and calculate V1, Vx2, 0 0 3 VA1,A1 and VA2,A2 .

5. (*) The linear transformation L: V → V has eigenvalues A1 and A2 such that A1 2. Recall for i = 1, 2 VA: := {v € V : L(v) = A;v} and define for i = 1,2 Vdg := {v € V : there is u e V, such that L(v) = d;v + u}. (This is called a generalised eigenspace – think of the vectors as "almost" eigenvectors, or eigenvectors up to an "error" term.) a) Prove that VA1A1 is a subspace of V and that VA, C VA1,d b) Prove that V1a O Va2 = {0}. 2 1 c) For A = 2 E Mat3x3(R), V = R³ and L = LA, find A1, 2 and calculate V1, Vx2, 0 0 3 VA1,A1 and VA2,A2 .

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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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.