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https://texercises.com/exercise/endomorphisms-eigenvalues-and-dimensions/
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Exercise:
Ase V is finite dimensional Tin textEndV and let lambda_...lambda_r be all the eigenvalues of T. Then T is diagonalizable iff _i^r textdim Eig_Tlambda_itextdimV.

Solution:
Longleftarrow: Choose a basis mathcalB_i for textEig_Tlambda_i. Recall that W:textEig_Tlambda_+...+textEig_Tlambda_r is a direct . Longrightarrow B_...B_r is a basis for W. Now Wsubseteq V is a linear subspace and textdimW_i^r |B_i|_i^rtextdim Eig_Tlambda_itextdimV Longrightarrow WV Longrightarrow V has a basis namely B_...B_r whose elements are eigenvectors of T. T is diagonalizable Longrightarrow marked as exercise
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Exercise:
Ase V is finite dimensional Tin textEndV and let lambda_...lambda_r be all the eigenvalues of T. Then T is diagonalizable iff _i^r textdim Eig_Tlambda_itextdimV.

Solution:
Longleftarrow: Choose a basis mathcalB_i for textEig_Tlambda_i. Recall that W:textEig_Tlambda_+...+textEig_Tlambda_r is a direct . Longrightarrow B_...B_r is a basis for W. Now Wsubseteq V is a linear subspace and textdimW_i^r |B_i|_i^rtextdim Eig_Tlambda_itextdimV Longrightarrow WV Longrightarrow V has a basis namely B_...B_r whose elements are eigenvectors of T. T is diagonalizable Longrightarrow marked as exercise
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Tags
basis, eigenvalue, eigenvector, eth, fs23, lineare algebra, proof
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Difficulty
(3, default)
Points
0 (default)
Language
ENG (English)
Type
Proof
Creator rk
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