 Monograph

#### Degree

Master of Science

#### Program

Applied Mathematics

Jeffrey, David

#### Abstract

A Pythagorean vector is an integer vector having an integer 2-norm. Such vectors are closely related to Pythagorean n-tuples, since n-tuples are the building blocks for Pythagorean vectors. Pythagorean vectors are, in their turn, the building blocks for rational orthonormal matrices. The work in this thesis has a pedagogical application to the QR decomposition of matrices, widely used in Linear Algebra. A barrier for students learning the details of the QR decomposition of a given matrix A is the occurrence of square-roots that cannot be simplified during the application of the two standard algorithms, namely the Gram--Schmidt method and Householder transformations. This thesis studies Pythagorean vectors and their application to the construction of exercises and test questions in which a given matrix A can be factored into matrices Q and R, with all arithmetic operations resulting in rational quantities, free from square roots. This freedom from square roots applies to every step of the calculations, and not just the final result.

As a preliminary to QR decomposition, the thesis explores the properties of Pythagorean vectors, including their generation for an arbitrary specified dimension. Pythagorean triples, which correspond to Pythagorean vectors of dimension 2, have been widely and enthusiastically studied in the literature, but higher dimensions have been less studied, and this thesis adds some new observations to previous studies.

#### Summary for Lay Audience

This thesis is based on a pedagogical application, namely the teaching of a particular topic in Linear Algebra. Courses in advanced linear algebra include a study of a process called the QR decomposition of a given matrix. The existing textbook treatments usually require students to perform extensive arithmetic operations. Matters can become more difficult for students when their calculations throw up awkward arithmetic expressions containing radicals, such as sqrt{168}, which cannot be simplified and which pollute the students' working. This thesis investigates ways in which instructors may construct exercises which are guaranteed to avoid unnecessary arithmetic difficulties for students.

In addition, Pythagorean triples are a popular subject for investigations in the literature. The thesis starts by happily joining this activity with new observations on the properties of triples, and adds observations of n-tuples for n>3.

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