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