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7) Suppose T E L(V) is diagonalizable. Prove that V = null T' range T.

7) Suppose T E L(V) is diagonalizable. Prove that V = null T' range T.

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter7: Eigenvalues And Eigenvectors
Section7.1: Eigenvalues And Eigenvectors
Problem 8E
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{p(T)|p e F[x]} # L(V). 7) Suppose T E L(V) is diagonalizable. Prove that V = null T O range T. 8) Suppose V is finite-dimensional, T E L(V) had dim V distinct eigenvalues, and S e L(V) has the same eigenvectors as T (not necessarily with the same eigenvalues). Prove that
Transcribed Image Text:{p(T)|p e F[x]} # L(V). 7) Suppose T E L(V) is diagonalizable. Prove that V = null T O range T. 8) Suppose V is finite-dimensional, T E L(V) had dim V distinct eigenvalues, and S e L(V) has the same eigenvectors as T (not necessarily with the same eigenvalues). Prove that
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