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https://texercises.com/exercise/characteristic-polynomial-and-endomorphisms/
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Exercise:
If V is an n-dimensional vector space over K and Tin textEndV then P_Tx is a polynomial of degree n with leading coefficient -^n.

Solution:
Proof. Let mathcalB be a basis for V. Then T-x id_V_mathcalB^mathcalB T_mathcalB^mathcalB-x I_n Longrightarrow P_TxtextdetT-xid_VtextdetT_mathcalB^mathcalB-x I_n P_T_mathcalB^mathcalBx. The claim of the remark now follows from a of the lemma shown before.
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Exercise:
If V is an n-dimensional vector space over K and Tin textEndV then P_Tx is a polynomial of degree n with leading coefficient -^n.

Solution:
Proof. Let mathcalB be a basis for V. Then T-x id_V_mathcalB^mathcalB T_mathcalB^mathcalB-x I_n Longrightarrow P_TxtextdetT-xid_VtextdetT_mathcalB^mathcalB-x I_n P_T_mathcalB^mathcalBx. The claim of the remark now follows from a of the lemma shown before.
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Tags
characteristic polynomial, endomorphism, 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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