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A 2020 Vision of Linear Algebra

ocw.mit.edu

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Re: A 2020 Vision of Linear Algebra

#51

Post by dntbnmpls »

brmgb wrote: post #36

After watching this and having read the comments, I am quite puzzled by the approach American seem to take to linear algebra. Are matrices viewed as the core of the subject in the USA ? My country curriculum introduces linear algebra through group theory and vector spaces. Matrices come later.

> After watching this and having read the comment, I am quite puzzled by the approach American seems to take to linear algebra. Are matrices viewed as the core of the subject in the USA ?

The US is a very big place. I doubt there is an american approach to linear algebra. We really don't have a single approach to anything. Different schools and majors probably approach the topic differently. My college had a linear algebra course specifically crafted for CS majors and engineers. I took that and it did focus on matrices. It was also the only math class that required programming. I believe math majors had their own linear algebra course.

> My country curriculum introduces linear algebra through group theory and vector spaces. Matrices come later.

Different strokes for different folks. If it worked out for you that's all that matters.

Re: A 2020 Vision of Linear Algebra

#52

Post by wegs »

arcturus17 wrote: post #9
willbw wrote: post #5

Earlier quoted context omitted.

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.

OCW opens up top-notch education to anyone and everyone, regardless of social or economic background. I wouldn’t take it for granted even with MIT’s eye-watering endowment, and I doubt donations to it are going to be paying cafeteria lunches for the students. I hope you’re donating and actively contributing to many non-profit projects and that your comment comes from being tired of the world’s injustices rather than…

Most money which comes into MIT passes through overhead. That means if a foundation donates to MIT, a bit over 1/3 of that money might ends up with whatever they donated to. A bit under 2/3 might go into the general budget (overheads vary by funding source, but the numbers above are from one specific project).

On paper, overhead is used for costs of running the place. In practice, it's used for things like upscale faculty clubs, million-dollar executive salaries, $200 million buildings, etc. MIT has among the highest overheads in the academy. Ironically, MIT claims its ocean yacht makes money rather than losing money (which could very well be true).

If you're okay with the majority of your money going to graft, donate to MIT. With a project like OCW, which has such a huge cost:benefit ratio, accepting the graft with the donation may be a rational decision, if you subscribe to a system of ethics like utilitarianism.

Personally, I almost never donate to a charity where the highest-earner makes more than I do. I think if everyone did that, MIT might lose some of the graft and corruption which has built up there over the years.

Re: A 2020 Vision of Linear Algebra

#53

Post by auggierose »

jacobolus wrote: post #45
auggierose wrote: post #23

Have not watched the videos yet, but that seems to me more like an 1820 vision of linear algebra :-) If you look at the order of topics in his book "An Introduction to Linear Algebra", you will find the topic "Linear Transformation" way back in chapter 8! Even after the chapters eigenvalue decomposition and singular value decomposition. But understanding that a matrix is just the representation of a linear transforma…

Gaussian Elimination is indeed from the 1820s. All the rest is more recent than that. The idea of matrix decomposition per se comes from the 1850s. The earliest work on something like the SVD is from the 1870s. You are onto something though. Strang is coming from a direction of numerical computations and algorithms for solving real-world problems. Pure mathematics departments for at least the past maybe 80 years ofte…

I believe in applications and theory going hand in hand together and benefiting each other. The computer is an incredibly powerful tool perfectly suited for this purpose. If we resist the urge to just see it as a push-button technology. Viewing matrices as a box of numbers instead of as a representation of a linear transformation leans too much in the direction of push-button for my taste.

Re: A 2020 Vision of Linear Algebra

#54

Post by jacobolus »

auggierose wrote: post #53
jacobolus wrote: post #45

Earlier quoted context omitted.

Gaussian Elimination is indeed from the 1820s. All the rest is more recent than that. The idea of matrix decomposition per se comes from the 1850s. The earliest work on something like the SVD is from the 1870s. You are onto something though. Strang is coming from a direction of numerical computations and algorithms for solving real-world problems. Pure mathematics departments for at least the past maybe 80 years ofte…

I believe in applications and theory going hand in hand together and benefiting each other. The computer is an incredibly powerful tool perfectly suited for this purpose. If we resist the urge to just see it as a push-button technology. Viewing matrices as a box of numbers instead of as a representation of a linear transformation leans too much in the direction of push-button for my taste.

Gil Strang does not view matrices as “just boxes of numbers”, nor does he teach that view.

YMMV, but I find pure mathematicians treat computers as “push-button technology” much more than applied mathematicians.

Re: A 2020 Vision of Linear Algebra

#55

Post by auggierose »

jacobolus wrote: post #54
auggierose wrote: post #53

Earlier quoted context omitted.

I believe in applications and theory going hand in hand together and benefiting each other. The computer is an incredibly powerful tool perfectly suited for this purpose. If we resist the urge to just see it as a push-button technology. Viewing matrices as a box of numbers instead of as a representation of a linear transformation leans too much in the direction of push-button for my taste.

Gil Strang does not view matrices as “just boxes of numbers”, nor does he teach that view. YMMV, but I find pure mathematicians treat computers as “push-button technology” much more than applied mathematicians.

I am not disagreeing with you there when it comes to pure mathematicians :-D

Edit: But there are of course big exceptions there as well, for example Thomas Hales.

Re: A 2020 Vision of Linear Algebra

#56

Post by andy_wrote »

tadhgds wrote: post #40
brmgb wrote: post #36

After watching this and having read the comments, I am quite puzzled by the approach American seem to take to linear algebra. Are matrices viewed as the core of the subject in the USA ? My country curriculum introduces linear algebra through group theory and vector spaces. Matrices come later.

I can't say whether or not it is the standard approach but I do know that it is very common in many countries to teach a linear algebra course that is heavy on matrix operations, that you can come away believing that linear algebra is somehow _about_ matrices and their operations. I know many in my university class seemed to believe that. A book I enjoyed is Axler's Linear Algebra Done Right[0], in which, if I rememb…

I've recently started going through Axler carefully and doing the problems, a quarantine activity I guess, and have been enjoying it. I actually learned about this book on an older HN post.

It does have plenty of matrices. The main thing it really does is avoid determinants until the very end. The determinant is certainly something I remember learning as a kind of rote operation, without really understanding any intuition behind why you'd multiply and add these numbers in this particular way. I still feel lacking in "feel" here, which is why I suppose I'm going through Axler now.

Re: A 2020 Vision of Linear Algebra

#57

Post by JoeCamel »

auggierose wrote: post #23

Have not watched the videos yet, but that seems to me more like an 1820 vision of linear algebra :-) If you look at the order of topics in his book "An Introduction to Linear Algebra", you will find the topic "Linear Transformation" way back in chapter 8! Even after the chapters eigenvalue decomposition and singular value decomposition. But understanding that a matrix is just the representation of a linear transforma…

I also think that this point should be emphasized more. It helped me a lot when I realized it. I also liked the abstract approach to vector spaces. Of course, a matrix is just one way to represent linear transformations and it can also represent other things like a system of linear equations.

Re: A 2020 Vision of Linear Algebra

#58

Post by T-zex »

srean wrote: post #24

I am familiar with the material of linear algebra but haven't read his books. Could someone who has absorbed linear algebra from different sources and familiar with Strang's books comment on what's good and bad and unique about them. In my time I had picked LA from Ben Noble, Halmos and Axler and the computation side of things from Golub & van Loan.

I'm still a beginner and I think Strang's Linear algebra books are more like a supplement material to his lectures. If you need to build a solid theoretical foundation of linear algebra you'd need to consider other resources too.

Having said that, he is explaining many things really well and is helping a lot to build intuition. He is always cautious presenting things that are computationally inefficient and suggests the alternatives.

Exercises are too hard for me personally. I'd prefer a more laborious set of exercises helping to cement the material, (as in calculus or usual algebra) and then have one or two problem solving puzzles at the end.

Re: A 2020 Vision of Linear Algebra

#59

Post by tchaffee »

auggierose wrote: post #23

Have not watched the videos yet, but that seems to me more like an 1820 vision of linear algebra :-) If you look at the order of topics in his book "An Introduction to Linear Algebra", you will find the topic "Linear Transformation" way back in chapter 8! Even after the chapters eigenvalue decomposition and singular value decomposition. But understanding that a matrix is just the representation of a linear transforma…

> Have not watched the videos yet,

Then please do. I took several online linear algebra courses from sources I trust and they were pretty bad. Or let's put it another way: I'm a pretty clever guy and I was still left confused. Strang is excellent in the classroom, and I almost even like videos for learning now thanks to him (x1.5 speed is your friend). His videos should not be your only learning source, but judging his course only by the book might result in a lot of learners skipping what I found to be the best course by far. If you want to learn linear algebra, give Strang a try first and you might save a lot of time.

Re: A 2020 Vision of Linear Algebra

#60

Post by srean »

T-zex wrote: post #58
srean wrote: post #24

I am familiar with the material of linear algebra but haven't read his books. Could someone who has absorbed linear algebra from different sources and familiar with Strang's books comment on what's good and bad and unique about them. In my time I had picked LA from Ben Noble, Halmos and Axler and the computation side of things from Golub & van Loan.

I'm still a beginner and I think Strang's Linear algebra books are more like a supplement material to his lectures. If you need to build a solid theoretical foundation of linear algebra you'd need to consider other resources too. Having said that, he is explaining many things really well and is helping a lot to build intuition. He is always cautious presenting things that are computationally inefficient and suggests…

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.

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