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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.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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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