11 ms·

Interactive Linear Algebra

981 points by kondor 7y ago

melling 7y ago
Lex Friedman interviews Gilbert Strang this week: https://youtu.be/lEZPfmGCEk0 https://youtu.be/lEZPfmGCEk0 Gilbert’s Linear Algebra course: https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/ https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra...
Zhyl 7y ago
Reminds me of the Nicki Case tool: https://ncase.me/matrix/ https://ncase.me/matrix/
bhntr3 7y ago
So far I think I prefer the visualizations in http://immersivemath.com/ila/index.html http://immersivemath.com/ila/index.html For pure visual intuition, I prefer 3blue1brown's Essence of Linear Algebra on Youtube (https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2x...). Even though they're not interactive, the visualizations themselves are the most compelling and insightful. Has anyone used both interactive texts? My initial impression is that the GA Tech text starts with matrices and applications to systems of equations (like Strang) while the immersive math one seems to start more focused on vectors and geometry with computer graphics applications. Neither seems to have supporting exercises yet which I think really limits their use as primary texts.
- sh-run 7y ago
  I took linear algebra after college (I hold a non-technical bachelors) through a local community college. 3blue1brown is what made linear algebra really click for me. My professor also gave us some jupyter notebooks to play around with that really helped. Unfortunately he developed these himself and I don't believe they are publicly accessible. I also purchased 'Coding the Matrix' by Philip N. Klein, which gave me another perspective on Linear Algebra. I primarily used it as an additional reference. I've been meaning to do a front to back reading of the text, but haven't gotten around to it yet. It's got some pretty decent exercises.
- dang 7y ago
  > So far I think I prefer the visualizations in http://immersivemath.com/ila/index.html http://immersivemath.com/ila/index.html A thread from 2015 about that book: https://news.ycombinator.com/item?id=10183725 https://news.ycombinator.com/item?id=10183725.
- impendia 7y ago
  > Neither seems to have supporting exercises yet I contacted one of the authors of the book. He confirms there aren't exercises yet, but says it's a work in progress and hopes there will be in the future.
tiborsaas 7y ago
I love how approachable the writing style is. It's very easy to write a technical introduction that assumes a lot of pre-existing knowledge of the reader. So far it's a great read for someone like me who got discouraged by notation and cloudy terms before. Looks like it's on pair with the Book of shaders.
- viburnum 7y ago
  It's insane how many math books are written assuming you already know what all the terms mean.
- paulgb 7y ago
  Link to Book of Shaders: https://thebookofshaders.com/ https://thebookofshaders.com/ (And a big +1 on the recommendation) Another piece of technical documentation that stuck out to me recently as remarkably readable (though I haven't finished reading it): https://craftinginterpreters.com/ https://craftinginterpreters.com/
- newnewpdro 7y ago
  The Book of Shaders is half vaporware, or am I missing something?
  - percentcer 7y ago
    yeah seems like they never finished it :( first half is great though
gautamcgoel 7y ago
I had one of the authors (Dan Margalit) as a prof when I was a math undergrad at Georgia Tech. Cool guy and a great instructor.
mamazaco 7y ago
This is our (my company) take on Linear Algebra, but then with exercises: https://app.bolster.academy/courses/chapters/en/6 https://app.bolster.academy/courses/chapters/en/6
kd0amg 7y ago
Linear: having to do with lines, planes, etc. This definition makes me worry that the book is going to operate entirely in R^n. Is there anything here about substantively different vector spaces?
- nostrebored 7y ago
  Abstract vector spaces is a follow up course to linear algebra at gatech
- impendia 7y ago
  Looking through the book, it seems to concentrate on R^n, and not pay much attention to abstract vector spaces. Probably a good choice for the intended audience (mostly engineers, likely). You might prefer Hefferon's Linear Algebra, also free and open-source, and which spends more time on this. http://joshua.smcvt.edu/linearalgebra/ http://joshua.smcvt.edu/linearalgebra/ Both books look truly excellent.
- deleted 7y ago
  [deleted]
- sitkack 7y ago
  Is it a real concern or are you humble bragging about the lack of advanced math in an introductory text on _linear_ algebra?
  - throwawaymath 7y ago
    No, that's a legitimate concern. It may or may not be relevant for what you specifically care about, but it's definitely a legitimate thing to ask about. It's not uncommon for even undergraduate linear algebra to cover abstract spaces and notions of linearity which generalize beyond R^n. For example, the function space P_n consisting of all polynomials with degree less than or equal to n. Hoffman-Kunze, Halmos, Axler and Friedberg-Insel-Spence are all examples of undergraduate textbooks which cover this material. This isn't just theoretical. Function spaces like P_n are useful in applied mathematics. And even if you don't use function spaces, it's very common for engineers, physicists and applied mathematicians to work in the complex space C^n rather than R^n.
impendia 7y ago
This looks outstanding. I am a math professor, and I need to choose a book next time I teach linear algebra. It may well be this one. If anyone has in-depth experience with the book, I'd be grateful to hear about it.
- qqaazz 7y ago
  It covers most of the standard material in an introductory Linear Algebra course. There is one large omission: the textbook completely ignores the notion of change of basis. There is no explanation why this was done, and a guess is that it was overlooked. The book is also very streamlined, and does not have much outside of the standard curriculum (which might be a plus).
  - cbolton 7y ago
    Change of basis is covered in section 5.3 on "similarity". See in particular the subsection "Geometry of Similar Matrices".
    - qqaazz 7y ago
      Aha. It appears it was just skipped in the curriculum rather, http://people.math.gatech.edu/~cjankowski3/19f/m1553/webpage/2019f-schedule.html http://people.math.gatech.edu/~cjankowski3/19f/m1553/webpage....
thinkingkong 7y ago
This is based on Mathbox which is... brilliant. Here's the demo blog post that shows off what it can do: https://acko.net/blog/mathbox2/ https://acko.net/blog/mathbox2/
- syntaxing 7y ago
  I normally skim through these types of articles but the visualization that s/he demos is so captivating and hypnotizing
  - romwell 7y ago
    Acko is Steven Wittens[1]; no need to use a pronoun when we have a name :) [1]https://acko.net/about https://acko.net/about
- knolan 7y ago
  This is great! Thank you. I’m rewriting fluid mechanics lecture notes in Jupyter (with Blender videos) and I’ve been looking for tools to visualise the maths. PyGel3D kinda works but it’s awkward and doesn’t work on nbviewer. I’m 100% on board with the death of PowerPoint and my students seem to agree now that they’ve seen Jupyter.
kernoble 7y ago
Awesome for reference and understanding. Anyone know of good practice material? Or maybe thinking of a way of generating practice material that exercises the component skills needed to do/understand Linear Algebra?
krosaen 7y ago
This is really neat, and it's clear a lot of work went into the polish and interactivity. I'd love to see the elimination examples extended to show how LU factorization falls out of those steps. It seems like a key thing to grasp and lays the groundwork for understanding of related matrix factorizations like symmetric, QR and singular value decomposition.
jacobsimon 7y ago
Wonderful! I remember struggling to make it through linear algebra in college because the teaching style was so unapproachable and disconnected from real applications. It wasn't until I took computer graphics that I really grasped a lot of the concepts.
majke 7y ago
From https://textbooks.math.gatech.edu/ila/overview.html https://textbooks.math.gatech.edu/ila/overview.html > Most engineering problems, no matter how complicated, can be reduced to linear algebra. Most problems... except a problem of too much linear algebra.
- black6 7y ago
  Everything is linear when plotted log-log with a fat magic marker.
lotaezenwa 7y ago
This extensively uses the American Institute of Math's knowls[0], which I think is excellent for math documentation. [0] https://aimath.org/knowlepedia/ https://aimath.org/knowlepedia/
Aardwolf 7y ago
Good stuff. One point of feedback: when making a widget fullscreen and then pressing the back button, you're back at the previous chapter, rather than scrolled at the position you left in the current chapter. I see now there's also a "make small again" button far down in the bottom right, but it's still a trap. Probably it's solved either by having the back button close the widget, or alternatively make the widget look more like an overlay and less like a new page (plus closing with "esc" would also help)
perl4ever 7y ago
I got an A- in linear algebra in college without learning anything, via multiple choice tests and pattern recognition. Recently, it's started to seem like a metaphor for the state of AI.
iLemming 7y ago
Also great resource: https://aiprobook.com/numerical-linear-algebra-for-programmers https://aiprobook.com/numerical-linear-algebra-for-programme...
Jahak 7y ago
Good job. Thank you very much.
s3arch 7y ago
Is there anything like this for other areas in mathematics such as Number Theory?
x7k 7y ago
Thank you
photon_lines 7y ago
If anyone is interested in a similar intuitive or geometric guide I open sourced my own notes and you can find them here: https://github.com/photonlines/Intuitive-Overview-of-Linear-Algebra-Fundamentals https://github.com/photonlines/Intuitive-Overview-of-Linear-...
kanibal 7y ago
lel

981 points by kondor 7y ago

Lex Friedman interviews Gilbert Strang this week: https://youtu.be/lEZPfmGCEk0 https://youtu.be/lEZPfmGCEk0 Gilbert’s Linear Algebra course: https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/ https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra...

Zhyl 7y ago

Reminds me of the Nicki Case tool: https://ncase.me/matrix/ https://ncase.me/matrix/

bhntr3 7y ago

So far I think I prefer the visualizations in http://immersivemath.com/ila/index.html http://immersivemath.com/ila/index.html For pure visual intuition, I prefer 3blue1brown's Essence of Linear Algebra on Youtube (https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2x...). Even though they're not interactive, the visualizations themselves are the most compelling and insightful. Has anyone used both interactive texts? My initial impression is that the GA Tech text starts with matrices and applications to systems of equations (like Strang) while the immersive math one seems to start more focused on vectors and geometry with computer graphics applications. Neither seems to have supporting exercises yet which I think really limits their use as primary texts.

sh-run 7y ago
I took linear algebra after college (I hold a non-technical bachelors) through a local community college. 3blue1brown is what made linear algebra really click for me. My professor also gave us some jupyter notebooks to play around with that really helped. Unfortunately he developed these himself and I don't believe they are publicly accessible. I also purchased 'Coding the Matrix' by Philip N. Klein, which gave me another perspective on Linear Algebra. I primarily used it as an additional reference. I've been meaning to do a front to back reading of the text, but haven't gotten around to it yet. It's got some pretty decent exercises.
dang 7y ago
> So far I think I prefer the visualizations in http://immersivemath.com/ila/index.html http://immersivemath.com/ila/index.html A thread from 2015 about that book: https://news.ycombinator.com/item?id=10183725 https://news.ycombinator.com/item?id=10183725.
impendia 7y ago
> Neither seems to have supporting exercises yet I contacted one of the authors of the book. He confirms there aren't exercises yet, but says it's a work in progress and hopes there will be in the future.

tiborsaas 7y ago

I love how approachable the writing style is. It's very easy to write a technical introduction that assumes a lot of pre-existing knowledge of the reader. So far it's a great read for someone like me who got discouraged by notation and cloudy terms before. Looks like it's on pair with the Book of shaders.

viburnum 7y ago
It's insane how many math books are written assuming you already know what all the terms mean.
paulgb 7y ago
Link to Book of Shaders: https://thebookofshaders.com/ https://thebookofshaders.com/ (And a big +1 on the recommendation) Another piece of technical documentation that stuck out to me recently as remarkably readable (though I haven't finished reading it): https://craftinginterpreters.com/ https://craftinginterpreters.com/
newnewpdro 7y ago
The Book of Shaders is half vaporware, or am I missing something?
- percentcer 7y ago
  yeah seems like they never finished it :( first half is great though

gautamcgoel 7y ago

I had one of the authors (Dan Margalit) as a prof when I was a math undergrad at Georgia Tech. Cool guy and a great instructor.

mamazaco 7y ago

This is our (my company) take on Linear Algebra, but then with exercises: https://app.bolster.academy/courses/chapters/en/6 https://app.bolster.academy/courses/chapters/en/6

kd0amg 7y ago

Linear: having to do with lines, planes, etc. This definition makes me worry that the book is going to operate entirely in R^n. Is there anything here about substantively different vector spaces?

nostrebored 7y ago
Abstract vector spaces is a follow up course to linear algebra at gatech
impendia 7y ago
Looking through the book, it seems to concentrate on R^n, and not pay much attention to abstract vector spaces. Probably a good choice for the intended audience (mostly engineers, likely). You might prefer Hefferon's Linear Algebra, also free and open-source, and which spends more time on this. http://joshua.smcvt.edu/linearalgebra/ http://joshua.smcvt.edu/linearalgebra/ Both books look truly excellent.
deleted 7y ago
[deleted]
sitkack 7y ago
Is it a real concern or are you humble bragging about the lack of advanced math in an introductory text on _linear_ algebra?
- throwawaymath 7y ago
  No, that's a legitimate concern. It may or may not be relevant for what you specifically care about, but it's definitely a legitimate thing to ask about. It's not uncommon for even undergraduate linear algebra to cover abstract spaces and notions of linearity which generalize beyond R^n. For example, the function space P_n consisting of all polynomials with degree less than or equal to n. Hoffman-Kunze, Halmos, Axler and Friedberg-Insel-Spence are all examples of undergraduate textbooks which cover this material. This isn't just theoretical. Function spaces like P_n are useful in applied mathematics. And even if you don't use function spaces, it's very common for engineers, physicists and applied mathematicians to work in the complex space C^n rather than R^n.

impendia 7y ago

This looks outstanding. I am a math professor, and I need to choose a book next time I teach linear algebra. It may well be this one. If anyone has in-depth experience with the book, I'd be grateful to hear about it.

qqaazz 7y ago
It covers most of the standard material in an introductory Linear Algebra course. There is one large omission: the textbook completely ignores the notion of change of basis. There is no explanation why this was done, and a guess is that it was overlooked. The book is also very streamlined, and does not have much outside of the standard curriculum (which might be a plus).
- cbolton 7y ago
  Change of basis is covered in section 5.3 on "similarity". See in particular the subsection "Geometry of Similar Matrices".
  - qqaazz 7y ago
    Aha. It appears it was just skipped in the curriculum rather, http://people.math.gatech.edu/~cjankowski3/19f/m1553/webpage/2019f-schedule.html http://people.math.gatech.edu/~cjankowski3/19f/m1553/webpage....

thinkingkong 7y ago

This is based on Mathbox which is... brilliant. Here's the demo blog post that shows off what it can do: https://acko.net/blog/mathbox2/ https://acko.net/blog/mathbox2/

syntaxing 7y ago
I normally skim through these types of articles but the visualization that s/he demos is so captivating and hypnotizing
- romwell 7y ago
  Acko is Steven Wittens[1]; no need to use a pronoun when we have a name :) [1]https://acko.net/about https://acko.net/about
knolan 7y ago
This is great! Thank you. I’m rewriting fluid mechanics lecture notes in Jupyter (with Blender videos) and I’ve been looking for tools to visualise the maths. PyGel3D kinda works but it’s awkward and doesn’t work on nbviewer. I’m 100% on board with the death of PowerPoint and my students seem to agree now that they’ve seen Jupyter.

kernoble 7y ago

Awesome for reference and understanding. Anyone know of good practice material? Or maybe thinking of a way of generating practice material that exercises the component skills needed to do/understand Linear Algebra?

krosaen 7y ago

This is really neat, and it's clear a lot of work went into the polish and interactivity. I'd love to see the elimination examples extended to show how LU factorization falls out of those steps. It seems like a key thing to grasp and lays the groundwork for understanding of related matrix factorizations like symmetric, QR and singular value decomposition.

jacobsimon 7y ago

Wonderful! I remember struggling to make it through linear algebra in college because the teaching style was so unapproachable and disconnected from real applications. It wasn't until I took computer graphics that I really grasped a lot of the concepts.

majke 7y ago

From https://textbooks.math.gatech.edu/ila/overview.html https://textbooks.math.gatech.edu/ila/overview.html > Most engineering problems, no matter how complicated, can be reduced to linear algebra. Most problems... except a problem of too much linear algebra.

black6 7y ago
Everything is linear when plotted log-log with a fat magic marker.

lotaezenwa 7y ago

This extensively uses the American Institute of Math's knowls[0], which I think is excellent for math documentation. [0] https://aimath.org/knowlepedia/ https://aimath.org/knowlepedia/

Aardwolf 7y ago

Good stuff. One point of feedback: when making a widget fullscreen and then pressing the back button, you're back at the previous chapter, rather than scrolled at the position you left in the current chapter. I see now there's also a "make small again" button far down in the bottom right, but it's still a trap. Probably it's solved either by having the back button close the widget, or alternatively make the widget look more like an overlay and less like a new page (plus closing with "esc" would also help)

perl4ever 7y ago

I got an A- in linear algebra in college without learning anything, via multiple choice tests and pattern recognition. Recently, it's started to seem like a metaphor for the state of AI.

iLemming 7y ago

Also great resource: https://aiprobook.com/numerical-linear-algebra-for-programmers https://aiprobook.com/numerical-linear-algebra-for-programme...

Jahak 7y ago

Good job. Thank you very much.

s3arch 7y ago

Is there anything like this for other areas in mathematics such as Number Theory?

x7k 7y ago

Thank you

photon_lines 7y ago

If anyone is interested in a similar intuitive or geometric guide I open sourced my own notes and you can find them here: https://github.com/photonlines/Intuitive-Overview-of-Linear-Algebra-Fundamentals https://github.com/photonlines/Intuitive-Overview-of-Linear-...

kanibal 7y ago

lel