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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

#31

Post by dalke »

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 in a paper. (In other words, don't feel like you should educate me on them here.)

Since you are a TA, don't you get some idea of where your students struggle with linear algebra?

Re: Immersive Linear Algebra – A free interactive online book

#32

Post by cabinpark »

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…

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'd calculus as well and I get similar responses. It's very frustrating sometimes.

Re: Immersive Linear Algebra – A free interactive online book

#33

Post by cabinpark »

ashishb4u wrote: post #27
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 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.

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.

Re: Immersive Linear Algebra – A free interactive online book

#34

Post by thyrsus »

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 corners and have an algorithm map the photo to its original rectangle - I intuit there's enough information in the photo and the four tagged points, given that the original is actually a rectangle.

Re: Immersive Linear Algebra – A free interactive online book

#35

Post by tiglionabbit »

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…

This exact application appears in the Coding the Matrix class on Coursera.

http://codingthematrix.com/

Re: Immersive Linear Algebra – A free interactive online book

#36

Post by dalke »

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…

I have TA'ed. Asking for questions in public doesn't help much. Students rarely like to admit they don't know something.

I was thinking more of when you look at their assignments. There are often multiple ways to approach a problem, and the route chosen can reveal something about one's level of comfort.

It doesn't help that university TAs get almost no training in how to be a TA. At least, I didn't.

Re: Immersive Linear Algebra – A free interactive online book

#37

Post by mcintyre1994 »

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…

Definitely doable - see the various scanner apps in your app store of choice. They work really well.

Re: Immersive Linear Algebra – A free interactive online book

#38

Post by neuronic »

dagw wrote: post #29
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 found that the level of abstraction that linear algebra is taught at is really weird. On the one hand it was never really taught in connection with algebra in general, or put into a wider mathematical context. On the other hand we were never really shown any real world used of linear algebra or how the concepts maps onto practical applications. It wasn't before years later when A) I'd taken more abstract algebra co…

I think what courses often fail at is to specifically mention that all of these things are abstractions invented by people to deal with a specific issue and then WHY IT HELPS solving this issue.

Whenever I studied Linear Algebra it was simply "here's a matrix, here is some algorithm, use it to get some number seemingly coming out of nowhere and just believe us that this number is what you really need."

Note that I am not immediately interested in mathematical proof of why this method is correct. I want a plain English explanation of what is going on but instead you usually just get a bunch of notation to digest.

Re: Immersive Linear Algebra – A free interactive online book

#39

Post by neuronic »

cabinpark wrote: post #33
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.

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 hard to develop intuition and a real understanding of the concept.

For man it's just there, you memorize it, you apply it, you take the result and simply have faith that it is what you need because some book/ prof said to solve problem X use eigenvalues.

Re: Immersive Linear Algebra – A free interactive online book

#40

Post by dalke »

wodenokoto wrote: post #25
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…

What it is, I think was my major problem. Matrix multiplication, ranks etc. are some of the most random seeming operations you can tell a student to memories.

My view is that the people who get to matrix algebra didn't have problem understanding the random seeming operation of previous math courses. Consider the existence of the "FOIL" mnemonic for multiplying two binomials (https://en.wikipedia.org/wiki/FOIL_method ), or SOH-CAH-TOA, where it was easier for me to just remember the definitions than the acronym. I think those are hints that some people consider those to be 'random seeming'.

When I took algebra, I remember being tripped by all the new vocabulary. Why do mathematicians say something is "Abelian" instead of 'commutative'? Why is it important to have terms like "group", "ring", and "vector space"? Sure, I learned the definitions, but at that level they were 'random seeming'.

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