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Root Preserving Transformations of Polynomials

by Branko Ćurgus (Western Washington University) and Vania Mascioni (Ball State Uniuversity)

This article originally appeared in:
Mathematics Magazine
April, 2007

Subject classification(s): Algebra and Number Theory | Linear Algebra | Linear Transformations

The article answers negatively the question, “Is there a (non-trivial) linear transformation \(T\) from \(P_n\), the vector space of all polynomials of degree at most \(n\), to \(P_n\) such that for each \(p\) in \( P_n\) with a real or complex root, the polynomials \(p\) and \(T( p)\) have a common root?" The proof is based on the fact polynomials of degree at most \(n\) have at most \(n\) roots in the real or complex numbers. This article investigates an area common to algebra and linear algebra.

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Capsule Course Topic(s):
Linear Algebra | Linear Transformation
Linear Algebra | Vector Spaces, Subspaces
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