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A 2020 Vision of Linear Algebra

ocw.mit.edu

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Re: A 2020 Vision of Linear Algebra

#71

Post by swiley »

vertak wrote: post #70

Can anyone grow why in the first 5 minutes of part 1 he shows a 3 by 3 matrix multiples by a 1 by 3 vector yet verbally he pulls out of no where this idea that if you have _two_ 1 by 3 vectors that pass through the origin then their linear combinations can be represented by a plane? The jump from the 3D to the 2D has me lost and I gave up

I don’t know which video you’re talking about but two non parallel vectors are enough to represent a plane, the normal will be their cross product.

Also, if you have 3 dimensional vectors you were always in 3D.

Re: A 2020 Vision of Linear Algebra

#72

Post by katzgrau »

I had very intelligent linear algebra professor in college but he was, in my opinion, a very poor communicator. I paid attention to lectures and stared at the text, but couldn't really understand the material. For the first part of a linear algebra course, students who don't mind blindly following mechanical processes for solving problems can do very well.

Unfortunately I'm one of those people who tends to reject the process until I understand why it works.

If it wasn't for Strang's thoughtful and sometimes even entertaining lectures via OCW, I probably would have failed the course. Instead, as the material became considerably more abstract and actually required understanding, I had my strongest exam scores. I didn't even pay attention in class. I finished with an A. Although my first exam was a 70/100, below the class average, the fact that I got an A overall suggests how poorly the rest of the class must have done on the latter material, where I felt my strongest thanks to the videos.

So anyway, thank you Gilbert Strang.

Re: A 2020 Vision of Linear Algebra

#73

Post by neutronicus »

vertak wrote: post #70

Can anyone grow why in the first 5 minutes of part 1 he shows a 3 by 3 matrix multiples by a 1 by 3 vector yet verbally he pulls out of no where this idea that if you have _two_ 1 by 3 vectors that pass through the origin then their linear combinations can be represented by a plane? The jump from the 3D to the 2D has me lost and I gave up

If you multiply the 1x3 vector by all scalars from -infinity to infinity you get all the points on a line.

If you do the same for another 1x3 vector, and it is not parallel to the first, you get all the points on a different line.

These two lines define a plane (and the cross product of the two vectors defines its normal vector)

Re: A 2020 Vision of Linear Algebra

#74

Post by layoutIfNeeded »

pengaru wrote: post #66

It's ridiculous how much random college-level linear algebra textbook material I stared at before things clicked in the course of just jumping in and exploring 3D graphics and writing my own 3D vector, matrix multiplication and 3D transform headers and using them in making some games in plain C. At some point it's like "Wait, is linear algebra really just about heaps of multiplication and addition? Like every dimensi…

Sounds like a case of You Can't Tell People Anything: http://habitatchronicles.com/2004/04/you-cant-tell-people-an...

Re: A 2020 Vision of Linear Algebra

#75

Post by john4532452 »

willbw wrote: post #5
knzhou wrote: post #2

Gilbert Strang's linear algebra course blew my mind back in high school, and I still use insights from it every day. Strang has a particular lecturing style where he approaches every topic several times, often beginning many lectures before the main treatment. At first I thought it was a bit confusing, but later I realized it helped build fluency, just like a language class. I'm really thankful to MIT OCW for putting…

Of course it is up to you what you do with your money but, they have a $17.5B endowment so there may be more needy causes if you were so inclined.

Malcom Gladwell made a similar point on his podcast http://revisionisthistory.com/episodes/06-my-little-hundred-...

Re: A 2020 Vision of Linear Algebra

#76

Post by swiley »

pengaru wrote: post #66

It's ridiculous how much random college-level linear algebra textbook material I stared at before things clicked in the course of just jumping in and exploring 3D graphics and writing my own 3D vector, matrix multiplication and 3D transform headers and using them in making some games in plain C. At some point it's like "Wait, is linear algebra really just about heaps of multiplication and addition? Like every dimensi…

I think the best thing anyone told me about linear algebra was that “matrices are just the coordinate form of a linear map.” So applying the map is equivalent to multiplying it’s matrix etc.

Re: A 2020 Vision of Linear Algebra

#77

Post by john4532452 »

gfxgirl wrote: post #25
roenxi wrote: post #11

It is interesting to compare this with 3Blue1Brown's linear algebra introduction on YouTube. He seems to have been the only mathematician who has actually mastered the medium; linear algebra lends itself very well to animations. The mathematicians don't understand how badly they need to animate some of these concepts.

3Blue1Brown's linear algebra animations were fun to watch but they did almost nothing for me except the basic fact that the "linear" part means lines. The rest was effectively preaching to the choir so those that already know linear algebra nodded their heads and idiots like me were still flummoxed

Yes. 3Blue1Brown has himself said multiple times no videos is substitute for text books.

Re: A 2020 Vision of Linear Algebra

#78

Post by srean »

T-zex wrote: post #67
srean wrote: post #60

Earlier quoted context omitted.

So the focus is on different recipes to cook a matrix with ? Different operations one can do on a matrix ? I hope its not just that, that would be very limiting considering what linear algebra is about and capable of.

No, his books are not recipes. The thing I'm struggling to communicate here is that he's got a more pragmatic style compared to other text books. The material he presents is complete and and he is doing great job making it approachable for non mathematicians. His books usually expand on the subjects he presents at his online lectures. I see them as advanced lecture notes.

No worries, not your fault, its really on me (to read the books). Its not really fair of me to ask for a comprehensive description in comments. Thanks for your comments anyhow.

Re: A 2020 Vision of Linear Algebra

#79

Post by DagAgren »

Phlogistique wrote: post #69
pengaru wrote: post #66

It's ridiculous how much random college-level linear algebra textbook material I stared at before things clicked in the course of just jumping in and exploring 3D graphics and writing my own 3D vector, matrix multiplication and 3D transform headers and using them in making some games in plain C. At some point it's like "Wait, is linear algebra really just about heaps of multiplication and addition? Like every dimensi…

Is it the same issue as the infamous "monads tutorials" problem, where the understanding takes a lot of time to infuse but looks obvious in retrospect when it finally clicks?

I am convinced that monads induce a very specific kind of brain damage that makes a person incapable of ever explaining monads.

Re: A 2020 Vision of Linear Algebra

#80

Post by bencw »

pengaru wrote: post #66

It's ridiculous how much random college-level linear algebra textbook material I stared at before things clicked in the course of just jumping in and exploring 3D graphics and writing my own 3D vector, matrix multiplication and 3D transform headers and using them in making some games in plain C. At some point it's like "Wait, is linear algebra really just about heaps of multiplication and addition? Like every dimensi…

Honestly, I think working with computers (possibly some programming) should be more frequently integrated into math courses. A computer is a natural to really interact with the material, like labs in the natural sciences. We're lucky to live in a time where this is possible, but sadly math education is taking it's sweet time taking advantage of this possibility.
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