San Siego: S. Gill Williamson, 2015. — 111 p.This material is a rewriting of notes handed out by me to beginning graduate students in seminars in combinatorial mathematics (Department of Mathematics, University of California San Diego). Topics covered in this seminar were in algebraic and algorithmic combinatorics. Solid skills in linear and multilinear algebra were required of students in these seminars - especially in algebraic combinatorics. I developed these notes to review the students’ undergraduate linear algebra and improve their proof skills. We focused on a careful development of the general matrix canonical forms as a training ground.Content:Functions and PermutationsAlgebraic terminology Sets, lists, multisets and functions Permutations ExercisesMatrices and Vector SpacesReview Exercises: subspaces Exercises: spanning sets and dimension Matrices – basic stuffDeterminantsElementary properties of determinants Laplace expansion theorem Cauchy-Binet theorem Exercises: Cauchy Binet and LaplaceHermite/echelon formsRow equivalence Hermite form, canonical forms and uniqueness Stabilizers of GL(n,K); column Hermite forms Finite dimensional vector spaces and Hermite forms Diagonal canonical forms – Smith formSimilarity and equivalenceDeterminantal divisors and related invariants Equivalence vs. similarity Characteristic matrices and polynomials Rational and Jordan canonical forms. Index
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