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A 2020 Vision of Linear Algebra

ocw.mit.edu

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Re: A 2020 Vision of Linear Algebra

#111

Post by ansible »

katzgrau wrote: post #72

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…

> 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.

I had a similar, though sort of opposite experience.

In high school, I breezed through the material, and started teaching myself calculus during the summer to prepare for university. Other than being a lazy student, I had no problems taking the 2nd semester advanced calc 2 and 3 courses my freshman year. I totally get what's being taught. There weren't a ton of practical examples, but I can easily see (for example) what the purpose of integration is, and how and why you'd do it in two or more dimensions. I could work the equations, no problem. Everything is great.

Along comes sophomore year, and still thinking I am hot stuff, I take advanced linear algebra and differential equations. More of the same, I thought.

Well... we seemed to spend the entire semester just solving different kinds of equations. No explanations given as to what they are for, where they are used, or what the point of any of it was. I struggled, for the very first time.

I either got a D or F for the mid-term exam, which was shocking to me.

We had one chapter where we were doing something practical. This is where you have a water tank, and a hole in to bottom. Because the pressure lessens as the tank empties, the flow rate is not constant. However, you can solve this via diff equations, and I really grokked it. I finally saw the point for some of what we had been doing. But it was just that one chapter, we skipped any other practical aspects for what we were studying.

I did end up pulling out a 'C' with that class, to my relief. Sure, most of the blame for my lousy performance must rest with me, because of my poor study habits. And a little blame can go to the TA, who wasn't a good communicator, so that hour every week was kind of useless. But I also blame the material and how it was presented.

Re: A 2020 Vision of Linear Algebra

#112

Post by [deleted] »

gowld wrote: post #101
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.

Mathematicians and teachers aren't all computer programmers, so creating animations isn't their forte. Books have pictures that do a pretty good job. Animations are pretty and interestit but that isn't the same as teaching all the math.

[deleted]

Re: A 2020 Vision of Linear Algebra

#113

Post by einpoklum »

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.

There's nothing particularly linear about groups.

Re: A 2020 Vision of Linear Algebra

#114

Post by samvher »

mangoman wrote: post #108
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…

Do you have any recommendations on videos/tutorials with graphics stuff that would help me grasp some of the linear algebra and basics about 2D/3D graphics? Your comment about 'staring at random college-level linear algebra' stuff resonated deeply with me, and I've always felt like I'm just not understanding how it all connects.

Not the person you responded to but I found this course to be very good: https://www.edx.org/course/computer-graphics-2

Re: A 2020 Vision of Linear Algebra

#115

Post by mattkrause »

wegs wrote: post #52
arcturus17 wrote: post #9

Earlier quoted context omitted.

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 fa…

MIT’s current overhead rate is 50.5%, but that’s pretty standard.

Here’s a scatterplot showing lots of research institutions’ rates. When this was published, MIT’s rate was slightly higher (54%). https://www.nature.com/news/indirect-costs-keeping-the-light... showing actual and calculated rates.

This also applies to federal research grants and is meant to cover costs associated with actually hosting the research (rent, utilities, support staff). Foundations can (and often do) negotiate lower rates. I’m not sure how donations are handled, but I don’t think the same F&A rates apply.

Re: A 2020 Vision of Linear Algebra

#116

Post by currymj »

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.

In general math departments in the US are less frightened of accidentally teaching the students something useful.

Re: A 2020 Vision of Linear Algebra

#117

Post by mattkrause »

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…

Indeed, Strang’s textbook starts with “I believe the teaching of linear algebra has become too abstract.”

He mentions this sentiment in a lot of interviews and things too.

Re: A 2020 Vision of Linear Algebra

#118

Post by jerf »

DagAgren wrote: post #110
jerf wrote: post #107

Earlier quoted context omitted.

The problem is telling people it's a container is "over describing" it. We don't need to hypothesize about that. We have the space suits and burritos to prove it is not a good didactic approach. It is not removing from the definition to simplify, it is adding to the definition, exactly as I carefully showed in my description of "Iterator". An Iterator is "a thing that presents a series of items". It does not simplify…

I'll just point out that neither of you have managed to really take a single step towards actually explaining monads.

I'm not trying, so that's not a surprise.

This has inspired me to try to update my post on the idea in a side window, but it's been sitting on my hard drive for over a year now and probably still has a ways to go yet.

Re: A 2020 Vision of Linear Algebra

#119

Post by plandis »

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…

When I was self studying linear algebra I found Strange’s book to be inadequate on its own, you really need the lectures to get the most value out of it.

I found going through Linear Algebra Done Right to provide a good counterbalance to Strang’s book+lectures.

Re: A 2020 Vision of Linear Algebra

#120

Post by rcthompson »

katzgrau wrote: post #72

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…

I had a very similar situation in my linear algebra course: in hindsight, I would literally have been better off teaching myself the material than listening to the professor. To this day it's still the main weak spot in my math/stats knowledge base. I'm really interested to check out these lectures.
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