https://www.youtube.com/playlist?list=PLdPQZLMHRjDK8ZbLIcq1Q...
Materials on Github:
Couldn’t agree more, Jack! Great times during 482… tranquil compared to the 470 slog that started immediately after every night :)
Great times. And I really liked how we did it all in C++ (other than computer vision 442 that was in matlab) rather than Python which some places do. Having that lower level understanding of languages in school makes understanding code so much easier, and something I didn't have to learn on my own.
For some reason linear algebra still isn't part of standard Mechanical Engineering course load (Calc 1, 2, 3, DiffEq) which made life extremely difficult in some of the later classes. I remember spending weeks brute forcing a lot of things that would have been trivial with a little bit of matrix math.
I took a superficially similar class as a 400 level elective but it assumed everyone already knew linear algebra going in, and it was a disaster.
https://ocw.mit.edu/courses/6-042j-mathematics-for-computer-...
and that assumption seems to be there as well, so very glad of the posting of the Youtube links elsethread.
Wow. In my undergrad all engineering majors had to take linear algebra (calc 3 was optional for computer engineering).
Michigan doesn't seem to require it as the College of Engineering core classes or as part of the BSME (checked because they're who this course is through):
https://me.engin.umich.edu/academics/undergrad/handbook/bach...
And my alma mater has a very similar progression.
And some upper level courses have a prerequisite.
But indeed it does seem a avoidable for many MechE majors.
Funny thread about UM engineering students avoiding taking UM math classes. https://www.reddit.com/r/uofm/comments/15w18gv/reminder_you_...
It's not that I can't do calculus, I took it in high school, and then again in my first go-round in CS. It's that I hate calculus. Not the subject itself, just the grinding away at problem sets.
I did a refresher in pre-calc, calc I, calc II & discrete mathematics during COVID at the local community college (was planning to finish the few credits I need for an actual CS BS) & I started calc III twice (but dropped both times). I even got a 4.0 on my first calc III exam (and this was an in-person class, so no online shenanigans).
I just have some kind of weird aversion to 3 dimensional calculus. I have convinced myself that I'm simply not smart enough to actually do the work. I understand it, I just get clammy with it.
Truth be told, maths are my kryptonite. Despite working with numbers all day every day for 30+ years, and writing a lot of software over the years (and not just CRUD, but games of all things), I am absolutely ashamed that I just can't seem to grok math with any rigor.
I have all the Stewart textbooks on my shelf, many textbooks from libgen (ones I've seen recommended on HN from people who went to much better universities than I attended), and I even work through problems a few hours per week. I just can't seem to make that leap from a guy who's "good with numbers" (from a layperson's perspective) to a guy who's good at math.
Maybe I need to break open one of my physics textbooks and actually use the calculus in an applied context and that will break whatever mental barrier I have (I've even watched all of the 3 blue 1 brown videos, countless youtube lectures, etc).
Do all the homework problems check the answer in the back of the book. You’ll make it.
In books like Stewart, staring at a theorem until you can write it's proof should trivialize most problems in the book.
If a method for solving a particular problem is too difficult for you maybe consider researching and/or inventing some new method to solve these problem. People created these methods in the first place because earlier methods were too tricky
Or just focus on work that doesn't require hundreds of hours to gain proficiency. As long as you have time every day to stop, think, and come up with an idea that solves a problem you won't become intellectually unfit.
I have Stewart (both the standard version and Early Transcendentals), and I also have a book from 1967 by Tom Apostol (the 2 volume set that covers single & multivariable calculus, linear algebra, a subtle introduction to differential equations and some probability as well).
My gut feeling is that I just don't know the correct way to study math in general. I have no problem doing the work. But it feels more like mechanical or algorithmic solving than it does like true understanding. There is a difference. I can't deconstruct a problem and think in the abstract to come up with a different method to solve it.
And there always seem to be some fundamental truth that I'm always missing. A part of a proof here or an axiom there that seems obvious to other people who study these subjects that I just don't "see".
It's incredibly frustrating, because deep down I know I have the aptitude for this stuff. I guess that most subjects have always been easy for me. I could ace exams without cracking the book (or just skimming).
Math is not like that. You need to read. And then re-read. And then do. And then do some more. And then go back and re-read again to see what you missed. And there's a lot of things that are between the lines, and if you're not following it, those things fall by the wayside.
I just need to learn how to learn math. I need to learn how to deconstruct notation and proofs to truly understand them. And there's no shortcut. It's grind and grind until it all becomes clear. That sort of thing is just difficult for me.
Gilbert Strang has a textbook, also more intuitive and applied. Free PDF provided by MIT. Sylvanus Thompson's book is recommended here, again, intuitive, applied.
Other comments here, 3 hours isn't enough, use Math Academy, nobody gets it on the first approach, all seem relevant. One of the textbooks recommended here says in the preface that it's for a second course in Linear Algebra. Analysis is just calculus the second (or third) go round, and it's said to be the hardest class in a math major.
I am in your boat, but about linear algebra instead of calculus. This is what I try to get myself over the hump.
It schedules everything for you including the review so you just have to keep showing up to do the work.
Even as little as 30 minutes per day done consistently for months will have you make tremendous progress.
And once you master multivariable calculus, fields like probability and machine learning will be unlocked for you.
Most of the time I've found that the deeper I plunge into abstraction in math I get rewarded with an extremely elegant formalism. Its like upgrading your weapon in dark souls, the early game enemies get one-shotted when you go back.
I don't think this has changed much (but absolutely should). I've watched in real time as Micron representatives reject mechanical engineers and prefer résumés from industrial engineers for design roles due to their superior grasp on linear algebra and statistics. I'm paraphrasing but "it's easier to teach an IE how to do FEA than it is to teach a mechanical engineer DOE and Weibull analysis".
Thankfully the companies I've worked for have done a really good job with advanced stats training.
Or use it in their courses and earn students that they need to learn it so succeed?
Education is secondary; this is job training! We need to crank out people ready to drop into Boeing's way of doing things!
FEA == Finite Element Analysis: advanced method of predicting the strength of a product via numerical simulation.
DOE == Design of Experiments: evaluation of how the outputs of a system change as you vary the inputs. At a high level, you build model of the system, then vary all the inputs through their entire range and to build a response surface of the output.
I'm currently doing a masters in IMSE--Industrial and Management Systems Engineering--and yeah it's changed since the 70s or whenever it had its real heyday (they come in waves).
The updated curriculum for undergrad is essentially the same as a mechanical engineering for the first two years, but as they wander off into advanced mechanics and fluids, IMSE students are doing time studies, factory design (FLOW!), and lots of stats and algorithms.
I've actually had the pleasure of getting to gripe to our school's Industrial Advisory Board which seems mostly full of Boeing people. They want to know if the curriculum serves the students well and I preach to them that, actually, if you spend 6 or 9 extra credits on proper software engineering that you've created a monster... but they don't listen. Some even get kind of offended because they think that a career in project management is a fine way to go about life (why go to engineering school?)
Sorry programming blows your mind? Perhaps that's why we need to teach it? I've done a lot of ERP integrations in my career and I'm not sure who they think is most qualified to do those sorts of things.
Graduate school definitely made up for lost time... LA was very front and center in the applied math courses.
If you can handle it, fabulous. If not, you're really in deep doo-doo. There did not seem to be a half-way to me. Astounding exercises, and also some are astoundingly hard.
Would highly recommend https://mathacademy.com/courses/linear-algebra or https://mathacademy.com/courses/mathematics-for-machine-lear...
I originally spent time working through practice problems from one of Strang's books, now really appreciate how systematic math academy is in assessing, building a custom curriculum, then doing spaced repetition.
there's a strang text on computational science that was much more my speed (less of the baby talk and repetitive manual arithmetic exercises) and i think that some of the revisions that came later (+ "learning with data") were better.
i did not find doing endless exercises of gaussian elimination or qr factorization by hand on small matrices to be all that enlightening.
this michigan course looks awesome!
I think this post (from a math academy employee) has a good argument for why these sorts of exercises are important. It's about basic arithmetic, but I think it applies to tedious things like performing gaussian elimination on small matrices as well.
https://www.justinmath.com/if-you-want-to-learn-algebra-you-...
I like to come at it from both angles - higher level with useful applications, and then lower level "I could maybe implement this if I had to" exercises. The latter are tedious, and hard to motivate effort for without the former. Ultimately, as the post argues, I agree that if you don't understand the lower level (tedious) operations, you will only get so far in your ability to apply LA.
See:
Bruner / Spiral Curriculum.
Ebbinghaus / Spacing effect
Hattie / Deep-surface-transfer learning
Chunking ("How People Learn" has a good copy on this)
Etc.
The way you do this is you take a course, and then you take more courses. After a few years, it all connects and makes sense. The first course, I find, is often best short, simplified, and applied. Once you get through that, you can go deeper.
Different angles are nice too. For linear algebra:
- Quantum computing
- Statistics and probability
- Machine learning
- Control theory
- Image processing
- Abstract algebra / groups / etc.
- Computer graphics
All come to mind.
On a mile-high level, this course seems ideal for a first pass. On a detailed level, I'm confused by some licensing issues.
At least that was my experience when I taught it. See https://bentilly.blogspot.com/2009/09/teaching-linear-algebr... for more detail on my experience.
One of the umich grad school prereqs for economics was linear algebra, and it was literally just that - pure math.
After trying a couple of courses and books, I liked it the most because it gives a pretty deep overview of the concepts, alongside the numpy and matlab code, which I found refreshing.
It's has good amount of proofs and has sections designed to build your intuition, which I really appreciated.
Not why I'm going to study it though, but yea, I might want to switch.
They have courses on linear algebra and mathematics for machine learning.
No certificates, but you can always demonstrate your mastery in interviews.
You can always build a project as well.
It is self-paced, so may not be what you're looking for, and it is expensive ($1250 if you have a BS already), but I seriously considered going this route before deciding to save big $$ and attend the local community college (which was actually a decent decision).
Program link: https://netmath.illinois.edu/
They offer 2 linear algebra courses, Math 257, which is Linear Algebra with Computer Applications (likely the "easy" applied version) and Math 416, Abstract Linear Algebra. Some of these Netmath courses do not have online lectures, but the Abstract LA course has video lectures from 2016.
From their site: "Math 416 is a rigorous, abstract treatment of linear algebra. Topics to be covered include vector spaces, linear transformations, eigenvalues and eigenvectors, diagonalizability, and inner product spaces. The course concludes with a brief introduction to the theory of canonical forms for matrices and linear transformations."
When I was investigating what to do in order to solidify my math credentials (still a work in progress), I knew UofI was a good school, and figured credit in one of their courses (online or not) would not be a terrible investment. At a bare minimum it wouldn't be belittled or untrusted like other online certificates might.
Plus the credit should transfer anywhere, if that's important.
Just be warned that this is literally the graduate level linear algebra course taken by mathematics majors. If you are looking for applications, this might not be it. On the other hand, if you are looking for a deep understanding of the fundamentals - I would say you found it.
I should have posted the Math 257 description too. It also has lectures online as well as a synchronous Zoom component:
Introductory course incorporating linear algebra concepts with computational tools, with real world applications to science, engineering and data science. Topics include linear equations, matrix operations, vector spaces, linear transformations, eigenvalues, eigenvectors, inner products and norms, orthogonality, linear regression, equilibrium, linear dynamical systems and the singular value decomposition.