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".

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.4: Linear Transformations

Problem 34EQ

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