9.6. SYMMETRIC FUNCTIONS
445
Proof of Theorem
9.6.6
.
Since the
m
of a fixed degree
d
form a basis of
the linear space
K
S
d
OEx
1
; : : : ; x
n
Ł
, it is immediate from the previous lemma
that the
of degree
d
also form a basis of
K
S
d
OEx
1
; : : : ; x
n
Ł
. Therefore, the
symmetric functions
of arbitrary degree are a basis for
K
S
OEx
1
; : : : ; x
n
Ł
.
Moreover, because the
m
can be written as
integer
linear combina
tions of the
, it follows that for any ring
A
, the ring of symmetric poly
nomials in
AOEx
1
; : : : ; x
n
Ł
equals
AOE
1
; : : : ;
n
Ł
.
n
Algorithm for expansion of symmetric polynomials in the elementary
symmetric polynomials.
The matrix
T
was convenient for showing
that the
form a linear basis of the vector space of symmetric polynomi
als. It is neither convenient nor necessary, however, to compute the matrix
and to invert it in order to expand symmetric polynomials as linear combi
nations of the
. This can be done by the following algorithm instead:
Let
p
D
P
a
ˇ
x
ˇ
be a homogeneous symmetric polynomial of degree
d
in variables
x
1
; : : : ; x
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 Fall '08
 EVERAGE
 Linear Algebra, Algebra, Vector Space, Elementary symmetric polynomial, Symmetric polynomial, symmetric polynomials

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