9) Suppose V is finite-dimensional and T E L(V). Let 1, ...,Am denote the distinct nonzero eigenvalues of T. Prove that dim E (2,, T) + + dim E (2m, T) < ... dim(range T)

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Question 9

or explain why there can be no such opera
6) Suppose V is a finite-dimensional with dim V > 1 and T E L(V). Prove that
{p(T)|p E F[x]} # L(V).
%3D
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
SEL(V) has the same eigenvectors as T (not necessarily with the same
eigenvalues). Prove that ST = TS.
%3D
9) Suppose V is finite-dimensional and T E L(V). Let 1, ...,Am denote the distinct
nonzero eigenvalues of T. Prove that dim E (1,,T) + … + dim E (Am, T) <
dim(range T)
10)Define T E L(R²) by T(x, y) = (41x + 7y,-20x + 74y). Verify that the basis of T
69
with respect to basis (1,4), (7,5) is )
0 46
Transcribed Image Text:or explain why there can be no such opera 6) Suppose V is a finite-dimensional with dim V > 1 and T E L(V). Prove that {p(T)|p E F[x]} # L(V). %3D 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 SEL(V) has the same eigenvectors as T (not necessarily with the same eigenvalues). Prove that ST = TS. %3D 9) Suppose V is finite-dimensional and T E L(V). Let 1, ...,Am denote the distinct nonzero eigenvalues of T. Prove that dim E (1,,T) + … + dim E (Am, T) < dim(range T) 10)Define T E L(R²) by T(x, y) = (41x + 7y,-20x + 74y). Verify that the basis of T 69 with respect to basis (1,4), (7,5) is ) 0 46
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