5. Suppose that V is a vector space over a field F and that B = {v1, V2, . .., Un} is a basis of V.Suppose that 0 :V →V is a linear transformation that satisfies0(v;) = vj + 2 mijVi(1)i=j+1for some mij e F.• Prove that 0 has exactly one eigenvalue and determine what that eigenvalue is.• Suppose V = R³. Find linear transformations 01 : V → V and 02 : V → V which satisfycondition (1) with respect to the standard basis of V such that• There is a basis of V consisting of eigenvectors of 01• There is not a basis of V consisting of eigenvectors of 02.

Answered: Image /qna-images/answer/a1e6323d-fcfa-4d94-8774-a270c8db1225.jpg

5. Suppose that V is a vector space over a field F and that B Suppose that 0 :V →V is a linear transformation that satisfies {v1, v2, ..., Vn} is a basis of V. 0(v;) = vj + > mijVi (1) i=j+1 for some mij e F. • Prove that 0 has exactly one eigenvalue and determine what that eigenvalue is.

5. Suppose that V is a vector space over a field F and that B Suppose that 0 :V →V is a linear transformation that satisfies {v1, v2, ..., Vn} is a basis of V. 0(v;) = vj + > mijVi (1) i=j+1 for some mij e F. • Prove that 0 has exactly one eigenvalue and determine what that eigenvalue is.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.6: The Matrix Of A Linear Transformation

Problem 26EQ

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Question

5. Suppose that V is a vector space over a field F and that B = {v1, V2, . .., Un} is a basis of V.
Suppose that 0 :V →V is a linear transformation that satisfies
0(v;) = vj + 2 mijVi
(1)
i=j+1
for some mij e F.
• Prove that 0 has exactly one eigenvalue and determine what that eigenvalue is.
• Suppose V = R³. Find linear transformations 01 : V → V and 02 : V → V which satisfy
condition (1) with respect to the standard basis of V such that
• There is a basis of V consisting of eigenvectors of 01
• There is not a basis of V consisting of eigenvectors of 02.

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