9. Let T be an operator on an n-dimensional vector space V over a field F. (a) If T has n distinct eigenvalues prove that T is diagonalizable. If T is diagonaziable prove that the characteristic polynomial of T splits over F. (c) Give an example of an operator T such that the characteristic polynomial of T splits but T is not diagonalizable.

9. Let T be an operator on an n-dimensional vector space V over a field F. (a) If T has n distinct eigenvalues prove that T is diagonalizable. If T is diagonaziable prove that the characteristic polynomial of T splits over F. (c) Give an example of an operator T such that the characteristic polynomial of T splits but T is not diagonalizable.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter7: Distance And Approximation

Section7.1: Inner Product Spaces

Problem 11AEXP

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Complete Example 2 by verifying that {1,x,x2,x3} is an orthonormal basis for P3 with the inner product p,q=a0b0+a1b1+a2b2+a3b3. An Orthonormal basis for P3. In P3, with the inner product p,q=a0b0+a1b1+a2b2+a3b3 The standard basis B={1,x,x2,x3} is orthonormal. The verification of this is left as an exercise See Exercise 17..
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