Representable Matroids and Linear Codes Carrie G. Rutherford June 19, 2008 “Do all those words mean the same thing?” gasped Milo. “Of course.” “Certainly.” “Precisely.” “Exactly.” “Yes,” they replied in order Norton Juster,The Phantom Tollbooth 1 Matroids - Motivation Matroids started out as structures that captured the fundamental properties of dependence that are common to graphs and matrices. However, they have a richness that makes them beautiful in their own right. Let C 1 ,C 2 ,...,C n be the columns of a matrix M . Any subset of these columns is either linearly independent or linearly dependent. Thus, linear dependence partitions the family of subsets of the columns of M into two classes. This is not an arbitrary partition - for example, the following two theorems must hold. • any subset of an independent set is independent. • consider two independent sets of columns; N p of p columns, and N p+1 of p + 1 columns. Then N p , together with some column in N p+1 \N p , is an independent set of columns. 1