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College Algebra Exam Review 435

College Algebra Exam Review 435 - 9.6 SYMMETRIC FUNCTIONS...

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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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