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https://texercises.com/exercise/minimal-polynomial/
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Exercise:
forall Ain M_ntimes n exists gin Kx s.t. gA.

Solution:
Proof. Consider I_n A A^ A^...A^rin M_ntimes nK *. Since M_ntimes nK is a vector space of dimension n^ then if we take r n^ then the elements in * cannot be linearly indepent. Longrightarrow exists a_a_...a_rin K s.t. a_ I_n+a_ A+a_ A^+...+a_r A^r.
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Exercise:
forall Ain M_ntimes n exists gin Kx s.t. gA.

Solution:
Proof. Consider I_n A A^ A^...A^rin M_ntimes nK *. Since M_ntimes nK is a vector space of dimension n^ then if we take r n^ then the elements in * cannot be linearly indepent. Longrightarrow exists a_a_...a_rin K s.t. a_ I_n+a_ A+a_ A^+...+a_r A^r.
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Tags
eigenvalue, eigenvector, eth, fs23, lineare algebra, matrices, proof
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Difficulty
(3, default)
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Language
ENG (English)
Type
Proof
Creator rk
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