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Unformatted text preview: 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...
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
 Fall '08
 EVERAGE
 Algebra

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