Immersive Linear Algebra (2016)

Primer looks like an interesting system - I'm on the mailing list, good luck!

And you include solutions for your exercises? This is fantastic! Do you have any plans to write something like this for functional analysis?

Perhaps your free book can be used to develop some evolving tool for the student. For example: the student take a test and then the book select the best exercises and topics for that student.

I notice that there is an option to buy your book, and perhaps there are other options to somehow get money to you. I was wondering if you felt comfortable revealing approximately how much money has come to you as a result of your (donated) labours?

It is very hard to feel confident in learning when I have no way to gauge if I am on the right track or not.

Thanks again.

It's also hard to focus on design, coding and content solo, and writing great, cohesive content tends to be a private endeavour. I'm not surprised to see immersivemath.com is written by a team.

I know (through fond memory!) your texts have some interactivity in hyperlinking to solutions and I found that very useful, but making it work must have taken time and been a bit of a headache. Extrapolate that to writing a webapp and it's easy to imagine you would never finish.

And yes, the words 'legendary' and 'classic' are direct compliments to you: your Lin Alg sits next to Spivak's Calculus on my bookshelf. They are both immortal.

The technological endurance of LaTeX is legendary, though

Compare Dirac's book on general relativity with one people actually read(To learn from) and you'll see why hand-holding and diagrams are helpful

Since I create educational projects (luckily, Quantum Game with Photons http://quantumgame.io/ was partially supported by an outreach grant; all other writings and stuff - not so much), I would love to see some way to support such endeavors.

What sequence of courses do I take on the site to get to calculus, starting from the beginning with the basics?

> Just call them their proper name and re-enforce this.

I'll tell you a story. When I learned math in college I was stuck by a Galois theory, I just was unable to understand it. So I decided to look different authors, to find some other approaches to explaining things. I found at least one textbook which I was forced to read from a very beginning, because it used a very different notation which I couldn't understand at all.

It was not the first time I have meet a new (for me) notations for things I already knew. I spent two more years learning math, and finally I understood one thing: there are no such a thing as "proper name" in a math. If you need to refer to a some thing, you need to define it. Math is all about your ability to name and to define things, to invent a math notation to write this things down, and to stick your mind to live inside universe which was built by your definitions, names and notations. It is doesn't matter then how original your notation is. All that matter is what results you can achieve.

From all the branches of math algebra is the most about different notations for the same thing. Just take a look at multiplicative and additive notations for groups. You can use either with any group, nothing matter just your convenience. If you like to use a plus sign and write "a+b" then use additive notation. It you prefer to use no sign and write "ab" then use multiplicative one. In algebra it is your respected right to use notation you like. But it comes with a cost: you need to respect other's right to use any notation they like.

However, it is ultimately a very limited linear format covering the topic only lightly.

I think Hoffman & Kunze is really the best textbook for the arrangement of material I'm talking about, but it's outdated at this point and its exposition is strictly less pedagogical than Axler. My vision for the "perfect" linear algebra textbook would be loosely based on Hoffman & Kunze, but replacing a lot of the theoretical exposition with Axler's exposition from Linear Algebra Done Right, then keeping (and even expanding) the mterial on Gaussian elimination and matrices. You could get a solid three-semester sequence of computational and proof-based linear algebra out of such a textbook, and you'd be all set to go to something grad-level like Roman.

Unfortunately it's also hard to judge a lot of linear algebra textbooks because the bifurcation between proof-based and computation-based linear algebra isn't as clean as it is in calculus. Calculus has its own separate course sequence in real/complex analysis, whereas linear algebra doesn't have a distinct name for the more rigorous coverage of the subject. So you have a lot of textbooks which choose one or the other thing, which then results in an over-emphasis on computational stuff in the first course and a complete de-emphasis on the motivation in the second course. When you are learning these back to back in distinct courses that's usually fine, but many first courses actually jump right to Axler despite the words of caution he writes in his own preface.

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1. Noam Elkies uses it when he teaches Harvard's Math 55a. See his list of comments and errata for the book: http://www.math.harvard.edu/~elkies/M55a.17/index.html

As an example, I want to learn how to deploy my models on the cloud but the mechanics of how computing on the cloud works is a depth trade-off. I wish I could learn all about distributed computing concepts but I find that I have neither the time or energy for it, at times.

I have seen the hierarchy of math presented like this:

1. In high school, you're taught how to compute arc length. You need to do nothing but calculation, and it doesn't matter why it works.

2. In undergrad, you're taught to prove the formula for computing arc length. Given the theorem statement and the requisite axioms, you can show that the formula works.

3. In grad school, you're taught to derive the formula for arc length from first principles. Given a set of axioms, you come up with the theorem and prove it from scratch.

4. In research, you're not taught anything. Instead you solve the questions, "How should I define arc length? Why does my definition of arc length matter, and where is it useful? What can I prove with it?"

Contrary to (quite a bit of) popular opinion, I would hold that a truly rich understanding of the theory can only happen in the context of the trigger for the theory. You don't need to understand the context of systems of linear equations, or why linearity is a useful concept, to understand vector spaces. But it's a lot easier if you have that context. Likewise you don't need to use Gaussian elimination to solve a lot of questions which ask you to prove something about a vector space or a linear map. But if you have that context, you can use it in your proofs instead of miring in towering heights of complexity.

Consider an art project, where you sit in a booth, and watch an interesting video drama. But why is it interesting? The video is a graph of segments, like an old "choose your own adventure" story. And there's an eye tracker watching you. So if you're interested in characters A and B, the story mutates to emphasize them.

Consider a one-on-one tutor on a good day. Noticing a student engaged and enthused, they might reorder content on the fly to leverage that. Might emphasize different aspects, and alter presentation, based on observed interests and weaknesses. Or consider working with a young child, watching them read word by word, noticing their where they hesitate and frown, probing for their thoughts, adjusting difficulty, providing a path of development.

What if that was math content rather than a drama or children's book? What if we could do this at scale? Eye tracking is just one tech coming in on the coattails of VR/AR. A setting for personalization AI is another.

What if saying "the best way to organize and present this topic in a textbook", becomes like saying "the Capital mandates, that every teacher of this grade, will today all teach the following lesson, by saying the following words, regardless of local context"? While not on a national level, that is a real thing.

What might it take to start encoding an adventure graph for linear algebra? The "oh, if you like this perspective on this topic, you might like this similar perspective on this other topic"?

Or if we don't have the tooling for that yet, can we start thinking about the tooling? Or fruitfully do something else now, in preparation for opportunity? Perhaps Kahn academy problems in more flavors, in a richer graph? ML-based textbook aggregation, synthesis and retheming? Perhaps it's all not ripe yet. But something else, maybe?

And Searle is an interesting argument for Griffith's perpsective. Searle rose to fame talking about stuff he knew how to do ("speech acts") and in his later years embarassed himself with bad criticisms of AI, as in the Chinese Room, his garbled continuation of the fallacy of the Philosophical Zombie.

If you don’t expect them to be potential future faculty, then by all means let other people hand you the problems and paradoxes unless you’re in contact with experiment.

This SIAM published Linear Algebra textbook makes a point about the scaffolding, too: http://matrixanalysis.com/Scaffolding.html

This is kind of my point - there actually isn't a whole lot more than Gaussian elimination at the core of linear algebra until you get to rings and modules. Of course I don't mean that in the sense that there aren't important definitions or topics. The theory is very rich. What I mean is that most theoretical questions you can ask about vector spaces and linear maps (or more abstractly, the linearity of a thing), are reducible to a matrix representation and Gaussian elimination.

You don't have to use Gaussian elimination, and I'm not arguing the course should become rote computation of matrices. There is a middle ground. For example if I hand you a linear map and ask you about the dimension of its range, you can identify a spanning set, refine that to a basis and you have your answer. You could also just take the rank of the matrix representing that linear map reduced to row echelon form. I don't think you lose much theory one way or the other.

But I could be wrong - if you have a specific example I'd be interested in hearing it. I've tried to declare front and center that I'm biased from learning linear algebra through Hoffman & Kunze. Halmos also grounded his (very theoretical) book in a generous use of matrices. Not covering matrices until later is a fairly recent development, and I think it's because (for reasons I've stated elsewhere) many professors don't distinguish between the rote computation of matrices and the rich theory of matrices.

To be more clear, what I was suggesting is to take the section on rank, toss that into the section on matrices, and then toss that into the section on Gaussian elimination (preceding vector spaces). That's useful because the rank of a coefficient matrix tells you the nature of the solution space of the corresponding system.

Then when you cover the rank-nullity theorem in the chapter on linear maps, you'll have the context of 1) what rank means in a matrix, and 2) what rank means in a linear map. That sets up a deeper understanding of how to verify the rank-nullity theorem for any linear map by finding the rank and the nullity using the basic and free variables of the reduced row echelon matrix.