This post will begin with some basic theory covered from Sheldon Axler's Third Edition Linear Algebra Done Right. It's assumed that the reader is familiar with linear maps, and the notion of a basis of a finite vector space. See the previous post "Matrix Multiplication" on this blog site from February 27th for a review.

The result of linear maps acting like matrix multiplication "... shows how the notions of the matrix of a linear map, the matrix of a vector, and matrix multiplication fit together" (pg 85). Formally stated, we have the theorem below.

Suppose \(T \in \mathcal{L}(V,W)\), \(v \in V\), and \(v_1,\ldots,v_n\) is a basis of \(V\) and \(w_1,\ldots,w_m\) is a basis of \(W\). Then \(\mathcal{M}(Tv)=\mathcal{M}(T)\mathcal{M}(v)\)

Axler makes clear about his book that it "... concentrate[s] on linear maps rather than matrices ... [and that] sometimes thinking of linear maps as matrices ... gives important insights that we will find useful." Later he states that the study of linear maps from a finite-dimensional vector space to itself constitute the most important part of linear algebra.

It's great when you can apply this basic theory to build and engineer a system of practical use. Currently the trend is to use linear algebra for building deep-learning models that use methods from data science. Deep learning and data science seem to be the fad topics in mathemtatics these days. In the preface to Gilbert Strang's Linear Algebra and Learning from Data, "... linear algebra is everywhere in the world of learning from data."

Mastering the concepts from Axler's book opens doors to not only better experiencing this kind of applied linear algebra in action, but also to a more engaging study of matroids. At that point in one's development, a more abstract treatment of geometry and combinatorics begins to reign on the mind. The Mathematics Magazine publication from Volume 8, Number 1, February 2009 has an article titled "Matroids You Have Known" for an introduction to that modern area of mathematics.