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OCTOBER 2000

**Row Reduction of a Matrix and ***A* = *CaB*

by Steven L. Lee and Gilbert Strang

slee@matrix.llnl.gov, gs@math.mit.edu

Every matrix has a unique reduced row echelon form *R* = rref (A). The *m*-by-*n* matrix *A* is reduced to *R* by a sequence of elementary row operations. The product of those operations is an elimination matrix *E* such that *EA* = *R.* Part of *E* is uniquely determined and part depends on the sequence of steps. Similarly, part of *E*-1 is determined and part is not. We show how *E* and *R* yield natural bases for the four fundamental subspaces (the column spaces and nullspaces of *A* and *A* ^{T}). They also produce a factorization of *A* into pivot columns times reduced rows. All of these bases and factorizations are conveniently produced by single commands in our MATLAB teaching codes, available at http://web.mit.edu/18.06.

This leads to the *echelon factorization A* = *CaB,* in which *B* contains the nonzero rows of rref(*A*), *C* contains the nonzero columns of (rref(A'))', of *A.* This factorization tells everything about the echelon bases for the fundamental subspaces. It was pointed out to us by Hans Schneider, and we learned that Albert Tucker and R. H. Bruck knew it too (but did not publish it).

**Constructing Kaleidoscopic Tiling Polygons in the Hyperbolic Plane**

by S. Allen Broughton

allen.broughton@rose-hulman.edu

Tiling the hyperbolic plane by iterated reflection in the sides of a kaleidoscopic polygon can be employed to make many beautiful and artistic patterns in the plane. These constructions may be greatly assisted by computer methods, provided we can construct the sides and vertices of these polygons, and hence the reflections in the sides of these polygons. We show how simple analytic geometry may be used to construct kaleidoscopic triangles and quadrilaterals. This work was motivated by work with undergraduates Dawn Haney, Lori McKeough, and Brandi Smith at the Rose-Hulman NSF-REU Tilings project site (http://www.tilings.org/index.html). Indeed, the methods of the paper have been used for visualization in a classification project of all divisible tilings of the hyperbolic plane. A description of the project is available at the same site. More importantly, figures depicting the divisible tilings in the classification?using the methods of this paper?are given in the file http://www.tilings.org/images/divquad/table_all.pdf.

**A Generalized Approach to the Fundamental Group**

by Daniel K. Biss

daniel@math.mit.edu

We begin with a description of a basic phenomenon of algebraic topology: the correspondence between subgroups of the fundamental group of a space *X* and connected covers of *X.* The essential theorems in this subject are given historical and intuitive motivation, but proofs are mostly omitted. In the second half of the paper, we summarize recent work that uncovers a new piece of structure in the fundamental group, and exploit that structure to demonstrate that the conditions under which the correspondence holds are actually far weaker than was once believed.

**The Sixtieth William Lowell Putnam Mathematical Competition**

by Leonard F. Klosinski, Gerald L. Alexanderson, and Loren C. Larson

**Putnam Trivia for the 90s**

by Joseph A. Gallian

jgallian@d.umn.edu

The annual Putnam competition has a long and glorious history of identifying extraordinary mathematical talent. Indeed, three Putnam Fellows (top 5 finishers) have won the Fields Medal and two have won a Nobel prize in Physics. In fact, the 1954 Harvard Putnam team included a future Fields Medalist and a future Nobel Laureate! And of course many Putnam Fellows have had distinguished careers starting with the very first winner of the Putnam Fellowship to Harvard?Irving Kaplansky. In the early days (1930s) only a few hundred students competed in the competition whereas by the 1990s over 2000 per year took part. In a 1989 issue of the MONTHLY I gave a list of trivia questions based on the first fifty years of the competition. In this article I offer trivia questions based on the competitions of the 1990s.

**NOTES**

**On Envy-Free Cake Division**

by Oleg Pikhurko

o.pikhurko@dpmms.cam.ac.uk

**Calculating Higher Derivatives of Inverses**

by Tom Apostol

apostol@caltech.edu

**The Modulus of Polynomials with Zeros at the Roots of Unity**

by Holly Carley and Xin Li

xli@math.ucf.edu, hkc7t@virginia.edu

**On the Norm of Idempotent Operators in a Hilbert Space**

by Vladimir Rakocevic

vrakoc@bankerinter.net

**A Simple Proof of a Theorem of Block and Hart**

by Guang Yuan Zhang

gyzhang@mail.tsinghua.edu.cn

**PROBLEMS AND SOLUTIONS**

**REVIEWS**

**The Interactive Geometry Software Cinderella**

By J. Richter-Gebert and U. H. Kortenkamp

Reviewed by H. Burgiel

burgiel@math.uic.edu

**The Nature of Mathematical Modeling**

By Neil Gershenfeld

Reviewed by Shirley B. Pomeranz

pomeranz@euler.mcs.utulsa.edu