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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

#51

Post by Jtsummers »

cabinpark wrote: post #32
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…

Interesting. I used linear algebra with quantum mechanics where eigenvectors represent quantum state of the Hamiltonian which is how I initially understood them. And as for TAing, have you ever TA'd? You definitely get a feeling but I can count numerous times when I've stood in front of the tutorial class and asked if there are any questions only to get no response back. I think it's a symptom of first years. I've TA…

> You definitely get a feeling but I can count numerous times when I've stood in front of the tutorial class and asked if there are any questions only to get no response back.

When I realized I was understanding subjects better than my classmates, I'd take on the task of asking the "dumb" questions for them. It was partly selfishness, I was tired of answering those questions for them outside of class. But it really did prove helpful to my classmates. It really helped that, when I was sitting with the students, I got to hear them mumbling and grumbling about what they didn't get. So I knew exactly what questions to ask to get the professor/TA to help my classmates.

I never TA'd myself so I have no idea if this would actually work, or at least work consistently. But, if you have a couple students that really seem to be getting the material, you could try talking to them one-on-one and ask them to help you out in this manner.

Re: Immersive Linear Algebra – A free interactive online book

#52

Post by wingerlang »

sudo_bang_bang wrote: post #12

I never studied math in college, in fact I barely studied math in high school. It was always daunting to try to parse the notation and figure out how the abstract symbols corresponded to some piece of reality. When I became a self-taught developer I found my math skills continuously lacking. I started teaching myself on Khan Academy and really picking it up a lot better because of the simplicity of the language and t…

What exactly do you do when you need math skills in developing? What do you develop with it? Some specific examples?

I've done math while developing 3D stuff, physics simulation and a lot of things - yet I never felt like my math skill mattered that much. And I really dislike theoretical math like in school.

Re: Immersive Linear Algebra – A free interactive online book

#54

Post by EvanKelly »

wingerlang wrote: post #52
sudo_bang_bang wrote: post #12

I never studied math in college, in fact I barely studied math in high school. It was always daunting to try to parse the notation and figure out how the abstract symbols corresponded to some piece of reality. When I became a self-taught developer I found my math skills continuously lacking. I started teaching myself on Khan Academy and really picking it up a lot better because of the simplicity of the language and t…

What exactly do you do when you need math skills in developing? What do you develop with it? Some specific examples? I've done math while developing 3D stuff, physics simulation and a lot of things - yet I never felt like my math skill mattered that much. And I really dislike theoretical math like in school.

Math is certainly not for everyone, but I actually found that once I started diving into pure math (NOT what was taught in my school's math curriculum) it was a whole new world.

Did your school actually teach pure math?

I bet a lot of programmer types had similar frustrations to the computationally intensive version of math taught in many high schools when they knew that these processes could simply be automated.

Re: Immersive Linear Algebra – A free interactive online book

#55

Post by something123 »

In terms of programming and linear algebra - please consult someone who is actually knowledgeable about the subject if you're implementing it in code.

Linear algebra without error analysis is very dangerous. Many many things are theoretically useful, but can't be used in practice. You can't calculate determinants, you can't count unique eigen values, you can't use certain decompositions.

Unfortunately this isn't really topic you can do a quick tutorial on and start writing new algorithms

Re: Immersive Linear Algebra – A free interactive online book

#56

Post by sudo_bang_bang »

wingerlang wrote: post #52
sudo_bang_bang wrote: post #12

I never studied math in college, in fact I barely studied math in high school. It was always daunting to try to parse the notation and figure out how the abstract symbols corresponded to some piece of reality. When I became a self-taught developer I found my math skills continuously lacking. I started teaching myself on Khan Academy and really picking it up a lot better because of the simplicity of the language and t…

What exactly do you do when you need math skills in developing? What do you develop with it? Some specific examples? I've done math while developing 3D stuff, physics simulation and a lot of things - yet I never felt like my math skill mattered that much. And I really dislike theoretical math like in school.

I haven't had to rely a lot on math skills for a large portion of my career. For most "full-stack" stuff I've been involved with, it barely gets any higher than algebra. However, I've found math creep up though here and there, whether it be in understanding how projections worked in the mapping software I wrote, the computer graphics code I assisted with, and also in my routing algorithms for traffic flow.

Nowadays my lack of knowledge is most apparent in studying machine learning. There have been some very high level concepts employed, at least for me: Jacobian matrices, partial derivatives, tensors, etc. I've been able to fake that for a while, the algorithms get abstracted in the libraries. Yet the abstraction obscures a real understanding for me in how they work.

Re: Immersive Linear Algebra – A free interactive online book

#58

Post by dougabug »

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…

The point of an eigenvector is that its direction is stable when you hit it with the given transformation. That is, it's an invariant direction of your transformation. This is analogous to the derivative of an exponential function being a simple multiple of that function. It makes it easier to reason about the action of the matrix, and simpler to represent.

Imagine I have a matrix transformation that takes a given input vector and converts it into a superposition of a thousand other vectors in random directions; that's far harder to reason about than if if just kicked it farther or brought it closer in the same direction.

Re: Immersive Linear Algebra – A free interactive online book

#59

Post by zornthewise »

neuronic wrote: post #39
cabinpark wrote: post #33

Earlier quoted context omitted.

Interesting. Eigenvalues and eigenvectors are one of the most fundamental applications of linear algebra. In physics, quantum mechanics is a giant eigenvalue-eigenvector problem. One application I really like is in machine learning: the eigenface algorithm.

That might all be the case. I would simply like a thoughtful introduction to the topic about where this abstraction comes from. What is the idea behind it and what was the motivation for its invention? After I understand the plain English concept, then give me the math notation and proof and applications. The way it works today is often to simply leave out the first part and I believe this is why many people find it…

The simple idea is that while linear maps(matrices) are generally a combination of rotations, scaling and other stuff in higher dimensions, sometimes they will act on a vector simply by scaling it.

Since scaling is a really easy operation to understand, the space generated by these vectors will be really easy to understand. Note that different vectors can be scaled by different amounts.

Now it often turns out that the space generated like this is actually the whole of the space under consideration and this really simplifies the linear map we started with. Hope that helps.

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