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Problem 5 Let R be the field of real numbers and {e1, 2} be the standard basis for R2. Consider the linear operator TE L(R²) defined by r(e₁) = 3e1 - e2, r(2)=4e1-202. Find the minimal polynomial for 7 and show that the rational canonical form for r is R= What are the elementary divisors of r? Hint: Given TEL(V), if m,(x) = av B:= aja +...+ak is the minimal polynomial for T th {U, T(U),...,7-¹(v)} is a basis for V and [r]B is the companion matrix of r and gir the minimal polynomial. Now, consider v=e₁

Problem 5 Let R be the field of real numbers and {e1, 2} be the standard basis for R2. Consider the linear operator TE L(R²) defined by r(e₁) = 3e1 - e2, r(2)=4e1-202. Find the minimal polynomial for 7 and show that the rational canonical form for r is R= What are the elementary divisors of r? Hint: Given TEL(V), if m,(x) = av B:= aja +...+ak is the minimal polynomial for T th {U, T(U),...,7-¹(v)} is a basis for V and [r]B is the companion matrix of r and gir the minimal polynomial. Now, consider v=e₁

Advanced Engineering Mathematics
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 R be the field of real numbers and {e1, 2} be the standard basis Problem 5 for R2. Consider the linear operator TE L(R²) defined by r(e₁) = 3e₁ - e2, 7(₂)=4e1-202- Find the minimal polynomial for 7 and show that the rational canonical form for 7 is T R -1 What are the elementary divisors of r? Hint: Given TEL(V), if m,(x) = av a₁x +...+ak is the minimal polynomial for T th B = {U, T(U),...,7-¹(v)} is a basis for V and [T]B is the companion matrix of r and gir the minimal polynomial. Now, consider v = e₁
Transcribed Image Text:Let R be the field of real numbers and {e1, 2} be the standard basis Problem 5 for R2. Consider the linear operator TE L(R²) defined by r(e₁) = 3e₁ - e2, 7(₂)=4e1-202- Find the minimal polynomial for 7 and show that the rational canonical form for 7 is T R -1 What are the elementary divisors of r? Hint: Given TEL(V), if m,(x) = av a₁x +...+ak is the minimal polynomial for T th B = {U, T(U),...,7-¹(v)} is a basis for V and [T]B is the companion matrix of r and gir the minimal polynomial. Now, consider v = e₁
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