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Lecture 23 - Addendum Andrei Antonenko April 2, 2003 1 Proofs of the main results from the lecture Let’s recall the deﬁnition from one of the previous lectures. Deﬁnition 1.1. Function f ( a 1 ,a 2 ,...,a m ) is called multilinear if it is linear in every argu- ment, i.e. for any i f ( a 1 ,...,a 0 i + a 00 i ,...,a m ) = f ( a 1 ,...,a 0 i ,...,a m ) + f ( a 1 ,...,a 00 i ,...,a m ) f ( a 1 ,...,λa i ,...,a m ) = λf ( a 1 ,...,a i ,...,a m ) . Deﬁnition 1.2. Multilinear function f ( a 1 ,a 2 ,...,a m ) is called alternating if it changes the sign after interchanging any 2 arguments, i.e. for any i and j f ( a 1 ,...,a i ,...,a j ,...,a m ) = - f ( a 1 ,...,a j ,...,a i ,...,a m ) . If f is alternating, then it is equal to 0 if any 2 arguments are equal. It is true, because if we interchange these 2 arguments, the function will not change, but from the other hand, it should change its sign. So, it is equal to 0. Now we’re able to formulate the main result about alternating multilinear functions. Theorem 1.3. For any c R in the vector space R n there exists the unique alternating multi- linear function f , such that f ( e 1 ,e 2 ,...,e n ) = c (1) (where e i ’s are rows with 1 on i -th place, and 0’s on all other places). Moreover, this function is equal to f ( a 1 ,a 2 ,...,a n ) = c · X all permutations of n elements σ sgn( σ ) a 1

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