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Immersive Linear Algebra – A free interactive online book

immersivemath.com

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Re: Immersive Linear Algebra – A free interactive online book

#41

Post by ashishb4u »

cousin_it wrote: post #30
ashishb4u wrote: post #27

Earlier quoted context omitted.

I think it's mostly because mathematics is taught as is with little or no attention to practical applications of the theory. For instance, till date, I could not see how eigen values/vectors are useful.

One nice application is that the eigenvectors of a stochastic matrix correspond to stationary states of the Markov chain. That leads naturally to a conversation about PageRank, which is super useful because it made billions of dollars for some people. Then you could talk about the use of eigenvectors for principal component analysis in data science, which is making a lot of money for other people right now. If these…

Thanks, the application to PageRank looks interesting. I shall resurrect my knowledge on eigenvectors and eigenvalues and make some sense out of it.

Re: Immersive Linear Algebra – A free interactive online book

#43

Post by jeffwass »

dalke wrote: post #31
cabinpark wrote: post #24

I always see linear algebra on HN and many people comment on how they never understood the subject. This makes me ask: what exactly is it that people don't get about linear algebra? What makes it appear to be a difficult subject? As someone who has used linear algebra almost every day in some form over the last decade, it's hard to get a perspective of what aspects are challenging to the beginner. And since I TA cour…

I see that chapter 10, on eigenvalues/eigenvectors, is "Coming soon!". Eigenvectors never made intuitive sense to me. As with the various decompositions (Cholesky, LU, etc), I could apply the math as algorithms to follow, but never got to the point where felt I could apply them to new problems. Then again, in practice, I've only needed eigenvectors once since college, and it was more a rote implementation described i…

Linear algebra (and most calculus) didn't really click for me until I needed to apply it to physics problems.

The general idea started to make sense in mechanics class when I could see matrices are convenient shorthand for solving multiple variables at once, and which behave like 'regular' variables when trying to manipulate them algebraically.

Eigenvalues and eigenvectors didn't make sense until quantum mechanics, where your 'operator' is effectively a matrix and your 'state' is a vector. The allowed observed values are the eigenvalues, and your final state after measurement is the corresponding eigenvector. Wave functions are then an extension of this from discrete space (useful for modelling spins for instance) to the continuum of position space.

Re: Immersive Linear Algebra – A free interactive online book

#44

Post by yardie »

cabinpark wrote: post #24

I always see linear algebra on HN and many people comment on how they never understood the subject. This makes me ask: what exactly is it that people don't get about linear algebra? What makes it appear to be a difficult subject? As someone who has used linear algebra almost every day in some form over the last decade, it's hard to get a perspective of what aspects are challenging to the beginner. And since I TA cour…

I took LA in university and I really wish they had offered it in high school or even middle school. It was a requirement for my major but not a requirement for any of my other math courses so I treated it like an elective. It was only after I took the course and started applying it that I realized how applicable it was to almost every math problem.

Instead of calculus for business majors or art majors they really should be using LA instead.

At the moment I'm trying to teach my son LA but I can't find any books for 8-10yo

Re: Immersive Linear Algebra – A free interactive online book

#46

Post by geomark »

dalke wrote: post #31
cabinpark wrote: post #24

I always see linear algebra on HN and many people comment on how they never understood the subject. This makes me ask: what exactly is it that people don't get about linear algebra? What makes it appear to be a difficult subject? As someone who has used linear algebra almost every day in some form over the last decade, it's hard to get a perspective of what aspects are challenging to the beginner. And since I TA cour…

I see that chapter 10, on eigenvalues/eigenvectors, is "Coming soon!". Eigenvectors never made intuitive sense to me. As with the various decompositions (Cholesky, LU, etc), I could apply the math as algorithms to follow, but never got to the point where felt I could apply them to new problems. Then again, in practice, I've only needed eigenvectors once since college, and it was more a rote implementation described i…

I enjoyed this articled called "Principal Components Analysis for Dummies" that illustrates eigenvectors and eigenvalues. https://georgemdallas.wordpress.com/2013/10/30/principal-com...

Re: Immersive Linear Algebra – A free interactive online book

#47

Post by gjm11 »

thyrsus wrote: post #34

At some point, I'm hoping to get enough of this to solve this problem: you've taken a picture of, say, the Mona Lisa in its rectangular frame, but because of crowds you weren't in line with dead center, instead you were 5 meters back, 1 meter high, and 2 meters to the side, and not even pointed at the center. Your photo now contains some quadrangle that is a projection of the rectangle. I'd like to tag the four corne…

A word of warning about solutions to this problem. The "easy" version states it this way: the mapping you want takes (x,y) to ((ax+by+c)/(gx+hy+i),(dx+ey+f)/(gx+hy+i)) for some a,b,c,d,e,f,g,h,i -- a general "projective transformation". If you decide ahead of time exactly what rectangle you want to map to, you have 8 constraints on a..i (two coordinates for each of four corners) and one irrelevant degree of freedom (if you multiply all of a..i by the same constant, nothing changes) so you have "exactly enough" information to find those parameters. So you solve the equations and you're done.

This works, provided your measured positions are perfectly accurate, but it isn't the best you can do. The reason is that a single projection actually has one degree of freedom less than a general projective transformation. (Three parameters to say where your camera is located. Three to say how it's rotated. One scaling factor. Total is 7 rather than 8.) This means that there's an extra (slightly hairy) constraint on the parameters a..i, which turns out to mean that the aspect ratio of the rectangle you're mapping to is determined for you.

Imposing that constraint will give you (1) more information -- you won't have to know ahead of time what aspect ratio you have -- and (2) a more accurately estimated transformation.

You can find more details of this stuff in, e.g., this PDF document: http://people.csail.mit.edu/bkph/articles/Harmful.pdf .

Re: Immersive Linear Algebra – A free interactive online book

#48

Post by gopowerranger »

cabinpark wrote: post #24

I always see linear algebra on HN and many people comment on how they never understood the subject. This makes me ask: what exactly is it that people don't get about linear algebra? What makes it appear to be a difficult subject? As someone who has used linear algebra almost every day in some form over the last decade, it's hard to get a perspective of what aspects are challenging to the beginner. And since I TA cour…

In almost every case, confusion arises when the "how" is taught with no explanation of the "why". Why does that formula work? Why did you use that formula?

Whole worlds were opened up to me when I learned that the fundamentals of trigonometry, like the simple computation of sin of an angle, are all related at its base to a square but no math teacher ever made that connection for me.

Re: Immersive Linear Algebra – A free interactive online book

#49

Post by Steuard »

jeffwass wrote: post #43
dalke wrote: post #31

Earlier quoted context omitted.

I see that chapter 10, on eigenvalues/eigenvectors, is "Coming soon!". Eigenvectors never made intuitive sense to me. As with the various decompositions (Cholesky, LU, etc), I could apply the math as algorithms to follow, but never got to the point where felt I could apply them to new problems. Then again, in practice, I've only needed eigenvectors once since college, and it was more a rote implementation described i…

Linear algebra (and most calculus) didn't really click for me until I needed to apply it to physics problems. The general idea started to make sense in mechanics class when I could see matrices are convenient shorthand for solving multiple variables at once, and which behave like 'regular' variables when trying to manipulate them algebraically. Eigenvalues and eigenvectors didn't make sense until quantum mechanics, w…

I love eigenvectors. For me, the rest of linear felt like a bunch of rote rules and algorithms for figuring out vaguely-interesting stuff. And then suddenly I find out that there's this amazingly non-obvious but remarkably powerful structure hidden inside matrices that I'd never even been aware of before.

Quantum mechanics is a huge application of eigenvectors, but I also really enjoy things like the "moment of inertia tensor", whose eigenvectors are the natural axes of rotation. Or better still, coupled oscillators, where the eigenvectors give "normal modes" of vibration. (And if you look at coupled first-order differential equations, eigenvectors can tell you all sorts of things about "trajectories" of the solutions. There are great applications of that to things like population dynamics in biology.)

Re: Immersive Linear Algebra – A free interactive online book

#50

Post by dagw »

cabinpark wrote: post #24

I always see linear algebra on HN and many people comment on how they never understood the subject. This makes me ask: what exactly is it that people don't get about linear algebra? What makes it appear to be a difficult subject? As someone who has used linear algebra almost every day in some form over the last decade, it's hard to get a perspective of what aspects are challenging to the beginner. And since I TA cour…

The other aspect that I found with LA compared to almost any other math subject I took was that it was almost entirely mechanical with little to no connection between my understanding of the subject and my grade. The exam questions where mostly along the lines of "find the eigenvector" and "find the determinant" rather than "what's an eigenvector" and "what's a determinant", so I got good grades despite not having any idea how to answer the latter questions, or really having any idea what I was doing. Understanding was neither necessary nor encouraged.
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