Wednesday, March 15, 2017

Linear Algebra - A YouTube Redemption

I bombed my first couple of Linear Algebra in-class quizzes, but learned from my mistakes and earned an A on the first test. On the midterm, I made some early mistakes that snowballed into panic. Silly algebraic errors, redoing problems more than once, using information from one question to answer another, not being able to answer a question. All the dummy moves (I'm not sure I want to even know my grade at this point).

This brought on some second guessing about whether I was really up to speed in class, and, just as important, if whatever knowledge gaps I might have might show up or impede upon an future learning. In order to make sure I had the concepts down, I spent the next week searching YouTube for videos of problems and concepts worked out from different perspectives. I hoped to find some combination that gave me an intuitive sense of what linear algebra was all about. What resulted can be found below.

Getting through all this will probably take me longer than the duration of my current course. But that's OK. The overall goal for studying math is as a foundation that allows me to learn other things that give me more insight into how the world works. For starters, I hope to understand the basics of calculus, statistics, logic, and linear algebra well enough so that when I encounter matrices, physical rates of change, summations, integrals, and statistical regressions in journal reports regarding local economies, energy resources and consumption, nature, urban infrastructure, biological traits, or social habits, I can read the material with the same sort of fluency I do my favorite literary and popular consumer magazines. Then be able to use that information to see deeper into the world around me as I walk through it.

Certainly, getting the feeling that my midterm exam grade could be any of the first four letters in the alphabet probably means I'm not quite yet on that trajectory. Even still, getting good grades isn't the same thing as having a sort of flexible fluency that allows you to absorb and utilize the language to see the world around you with an added degree of clarity. Either way, I can tell there's a bit of a gap in comprehension for me. And, if I'm reading things right, in a world of big data, digitally animated movies, games, and simulations, and policy driven by economic models, matrices and linear algebra will become an increasingly fundamental part of how society uses facts and figures to analyze the world around us. So it doesn't hurt to really take the time to try and learn this stuff.

The goal with this list is to have a sort of cascading effect that I can rinse, repeat, and rearrange if I need to. Each step adds to a foundation. Each next step makes use of previous material learned in the previous steps. If things are too easy, one can skip ahead. If they are too difficult, one can move backward. And even after being done with the class, I can continue to build on what I've learned.

Also, I suppose anyone who is as curious (or befuddled) as me might be able to use these resources as they see fit.

POTENTIAL PRIMER - My Why U will seem to cover really simple concepts. But I think it's important to know that you really know this stuff, before you even take a linear algebra class if possible. Some of the really simple things that caught me off guard early were being able to process and apply the methods of back-substitution or the visual and conceptual meanings of inconsistent vs. dependent vs. independent at an appropriate speed. I found myself stumbling over myself thinking too hard about "why's" and "how's" early on. Videos 39 - 51 is a good place to start. If you find you need to, check out earlier videos too. Anyway, this is what I'm doing.

1.)  START HERE -  Essence of Linear Algebra focuses on making concepts intuitive through visual means rather than computation. It's an effective and time-efficient way of getting the bigger picture in a relatively short amount of time. This video series won't give you the tools to actually solve the problems. But it make things clearer so that when you do, it won't just feel like rote calculations

2.) BASIC MATRIX OPERATIONS - After getting the concepts down, videos 52-61 of My Why U, followed by videos 03-1 to 03-4 of the Mth 309 video series can provide a good basic introduction to matrix operations and high level insight.

3) FOUNDATION IN THEORY AND PRACTICE - From what I can tell (I haven't gotten through all of the videos yet), the Khan Academy linear algebra series is both relatively thorough and comprehensive, but is both long in duration and requires a certain level of commitment. The series essentially explain a concept and proceeds to work through example problems for whatever is mentioned. 

4.) PROBLEM SOLVING - Understanding broad concepts and being able to work through problems aren't necessarily the same thing. Patrick JMT and Adam Panagos lean more heavily towards computation, problem-solving, and execution. Mathway is another great tool. That is, if you are willing to pay $20 a month for it.

5.) PUTTING IT ALL TOGETHER - Videos like The Big Picture of Linear Algebra and articles such as Better Explained: An Intuitive Guide to Linear Algebra are important, because they provide a quick way to check whether your understanding has evolved beyond just being able to work through problems, and into insight regarding how all the moving parts merge into a comprehensive whole. The real fun/interesting stuff though, is watching how the things you learn tie into the world at large (you don't actually need to know linear algebra to appreciate most of these videos):

If, at the end of all this, you find yourself wanting more regarding formal theory try Stepik 's linear algebra series.  Stepik takes a similar approach to Khan Academy, but uses denser language, formal mathematical notation, and proofs. Khan Academy will probably be a better fit for most people (including me). Stepik may be better if the concepts already come to you quickly; if you find Khan Academy isn't as precise or as you'd like to be; if you are taking a more thorough, proof-based linear algebra course; or if are planning to continue with classes like discrete math, proof writing, abstract algebra, or any material that includes any of the aforementioned subjects.