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https://texercises.com/exercise/linear-combination-of-n-linear-functions/
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Exercise:
Any linear combination of n-linear functions is again an n-linear function.

Solution:
Proof. Enough to prove for linear combination of two n-linear functions. Let DE:M_ntimes nKlongrightarrow K be n-linear function abin K. aD+bE:M_ntimes nK&longrightarrow K aD+bEA:a DA+bEA aD+bEalpha_...alpha_i+alpha_i'...alpha_n aDalpha_...alpha_i...alpha_n+Dalpha_...alpha_i'...alpha_n+bEalpha_...alpha_i...alpha_n+Ealpha_...alpha_i'...alpha_n aD+bEalpha_...alpha_i...alpha_n+aD+bEalpha_...alpha_i'...alpha_n. aD+bEalpha_...lambda alpha_i...alpha_n aDalpha_...lambda alpha_i...alpha_n+bEalpha_...lambda alpha_i...alpha_n lambda aD+bEalpha_...alpha_i...alpha_n.
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Exercise:
Any linear combination of n-linear functions is again an n-linear function.

Solution:
Proof. Enough to prove for linear combination of two n-linear functions. Let DE:M_ntimes nKlongrightarrow K be n-linear function abin K. aD+bE:M_ntimes nK&longrightarrow K aD+bEA:a DA+bEA aD+bEalpha_...alpha_i+alpha_i'...alpha_n aDalpha_...alpha_i...alpha_n+Dalpha_...alpha_i'...alpha_n+bEalpha_...alpha_i...alpha_n+Ealpha_...alpha_i'...alpha_n aD+bEalpha_...alpha_i...alpha_n+aD+bEalpha_...alpha_i'...alpha_n. aD+bEalpha_...lambda alpha_i...alpha_n aDalpha_...lambda alpha_i...alpha_n+bEalpha_...lambda alpha_i...alpha_n lambda aD+bEalpha_...alpha_i...alpha_n.
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eth, fs23, linear combination, lineare algebra, n-linear, proof
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ENG (English)
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Proof
Creator rk
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