Matrices are matroids

In Whitney's foundational paper [42], matroids were introduced as an abstraction of linear dependence. Throughout the history of the subject, connections between matrices and matroids have motivated an enormous amount of research. The first three questions we consider are fundamental:

1. Q1. Does every subset of vectors give rise to a matroid, i.e., do the subsets of linearly independent column vectors of a matrix always satisfy the independent set axioms? (Answer: Yes – Theorem 6.1.)

2. Q2. When do two different matrices give the same matroid? (Answer: Sometimes – Section 6.2.)

3. Q3. Does every matroid arise from the linear dependences of some collection of vectors? (Answer: No – Example 6.20.)

Concerning Q1, we've seen plenty of examples of matroids that come from specific matrices so far. In Chapter 1, matrices were the first examples we considered, and, in Chapter 3, we interpreted the matroid operations of deletion, contraction and duality for matrices. But, if you've been paying close attention, we never proved the subsets of linearly independent vectors satisfy the axioms (I1), (I2) and (I3). We fix that now by giving a proof that linear independence satisfies the matroid independence axioms.

Column dependences of a matrix

Theorem 6.1. Let E be the columns of a matrix A with entries in a field F, and let I be those subsets of E that are linearly independent. Then I is the family of independent sets of a matroid.