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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Transcribed Image Text:5. (*) The linear transformation L: V → V has eigenvalues A1 and A2 such that A1 2. Recall
for i = 1,2 VA; := {v E V : L(v) = A;v}
and define
for i = 1,2 VAd; := {v E V : there is u E V, such that L(v) = X;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 VA1,A1 is a subspace of V and that V, C V1,1.
b) Prove that Va1,a1 O Va2 = {0}.
2 1
c) For A =
2
E Mat3x3(R), V =
R³ and L = LA, find A1, A2 and calculate V, Va2,
0 0
3
VA1,A1 and VA2,Az.
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