8.C.7: Suppose V is a complex vector space. Suppose that PE£(V) is such that P2 =P. Prove that the characteristic polynomial of P is z"(z- 1)* where m= dim null P and k= dim range P. HINT: This problem has many moving parts, First, if P2 = P, what does that give you when you apply (8.5)? Second, use the previous decomposition to show that P can only have eigenvalues of zero and one. Third, remember that multiplicities must add-up to the dimension, and again use the decomposition from the first step. 8.5 V is the direct sum of null T dim V and range Tdim V Suppose T e L(V). Let n = dim V. Then V = null T" range T".
8.C.7: Suppose V is a complex vector space. Suppose that PE£(V) is such that P2 =P. Prove that the characteristic polynomial of P is z"(z- 1)* where m= dim null P and k= dim range P. HINT: This problem has many moving parts, First, if P2 = P, what does that give you when you apply (8.5)? Second, use the previous decomposition to show that P can only have eigenvalues of zero and one. Third, remember that multiplicities must add-up to the dimension, and again use the decomposition from the first step. 8.5 V is the direct sum of null T dim V and range Tdim V Suppose T e L(V). Let n = dim V. Then V = null T" range T".
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.3: Lines
Problem 22E
Related questions
Question
100%
From a
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:
9781305658004
Author:
Ron Larson
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:
9781305658004
Author:
Ron Larson
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning