Secolinsky.com latest post feedhttp://www.secolinsky.com/rss/The latest posten-usSat, 24 Aug 2019 08:27:49 -0000Linear Algebra for the Motivatedhttp://example.com/blogs/linear-algebra-basics,17/This post will begin with some basic theory covered from Sheldon Axler's <i>Third Edition <u>Linear Algebra Done Right</u></i>. 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 "<a href="http://www.secolinsky.com/blogs/matrix-multiplication,16/">Matrix Multiplication</a>" 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.
<span class="theorem">
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
\(\mathcal{M}(Tv)=\mathcal{M}(T)\mathcal{M}(v)\)
</span>
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 <u>Linear Algebra and Learning from Data</u>, "... linear algebra is everywhere in the world of learning from data."
---snippet---Mastering the concepts from Axler's book opens doors ---endsnippet---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 "<a href="https://www.maa.org/sites/default/files/pdf/shortcourse/2011/matroidsknown.pdf">Matroids You Have Known</a>" for an introduction to that modern area of mathematics. http://example.com/blogs/linear-algebra-basics,17/