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https://texercises.com/exercise/determinants-and-bases/
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Exercise:
Does textdetT_mathcalB^mathcalB dep on mathcalB?

Solution:
No. Let mathcalB' be another basis for V. Denote by P:id_V_mathcalB^mathcalB' the transition matrix between the basis mathcalB' and mathcalB. Recall that T_mathcalB^mathcalB' id_V_mathcalB'^mathcalB T_mathcalB^mathcalB' id_V_mathcalB^mathcalB' and also id_V_mathcalB'^mathcalBleftid_V_mathcalB^mathcalB'right^- So T_mathcalB'^mathcalB'P^- T_mathcalB^mathcalB PLongrightarrow textdetT_mathcalB'^mathcalB'textdetT_mathcalB^mathcalB.
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Exercise:
Does textdetT_mathcalB^mathcalB dep on mathcalB?

Solution:
No. Let mathcalB' be another basis for V. Denote by P:id_V_mathcalB^mathcalB' the transition matrix between the basis mathcalB' and mathcalB. Recall that T_mathcalB^mathcalB' id_V_mathcalB'^mathcalB T_mathcalB^mathcalB' id_V_mathcalB^mathcalB' and also id_V_mathcalB'^mathcalBleftid_V_mathcalB^mathcalB'right^- So T_mathcalB'^mathcalB'P^- T_mathcalB^mathcalB PLongrightarrow textdetT_mathcalB'^mathcalB'textdetT_mathcalB^mathcalB.
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Tags
basis, determinant, eth, fs23, lineare algebra, proof
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Difficulty
(3, default)
Points
0 (default)
Language
ENG (English)
Type
Proof
Creator rk
Decoration