# Polynomial Translation Groups

by Dan Kalman (University of Wisconsin)

This article originally appeared in:
Mathematics Magazine
January, 1983

Subject classification(s): Algebra and Number Theory | Linear Algebra | Matrix Algebra | Geometry and Topology | Plane Geometry | Transformations
Applicable Course(s): 3.8 Linear/Matrix Algebra

Consider the vector space of polynomials of degree less than $$n$$, and a polynomial $$p(x)$$ in this space. The author describes the matrix $$M(r)$$ that maps the polynomial $$p(x)$$ to $$p(x+r)$$, where $$r$$ is a real number. The group structure of the matrices $$M(r)$$ under multiplication then gives rise to various combinatorial identities.

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Capsule Course Topic(s):
Linear Algebra | Linear Transformation
Linear Algebra | Matrix Algebra
Linear Algebra | Matrix Multiplication