I had the fortune of taking 18.06 (Linear Algebra) from Gilbert Strang around the time of this essay. And like other commenters, I will agree that he was a very good teacher. I could tell that he held a great emotional attachment to matrices; he'd often anthropomorphize them with genders and refer to them as "little guys".
Too Much Calculus – Gilbert Strang (2001) [pdf]
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Re: Too Much Calculus – Gilbert Strang (2001) [pdf]
#92I think taking less calculus and more linear algebra is a mistake.
There is a really good book in two volumes that I feel shows how methods of linear algebra can be extended using calculus to solve a wide range of problems: A Course in Mathematics for Students of Physics:
http://www.goodreads.com/book/show/2685577-a-course-in-mathe...
Volume 2 is more important. It starts demonstrating how methods of Linear Algebra can be used to solve circuit problem, then generalizes this to higher dimensions and from discrete to continuous spaces using the exterior derivative. Seeing this demonstrated gave me a much clearer, more intuitive understanding for both calculus and linear algebra.
Re: Too Much Calculus – Gilbert Strang (2001) [pdf]
#93I've never understood why Calculus is taught as a separate course. It seems so detached from practicality. I've always thought students should take perhaps 4-6 semesters of Physics -- and introduce derivatives, integrals, etc. as needed when they can be demonstrated to more easily solve the physics problems. If the physics content was more spread out, there would be ample time to introduce the necessary math concepts, instead of stuffing them into a separate course.
Re: Too Much Calculus – Gilbert Strang (2001) [pdf]
#94I really can't agree with this strongly enough. I had enough Calculus and Differential Equations courses to kill a horse in college (Chemistry major) and I've never used that knowledge once in my entire programming career. But I find uses for linear algebra and matrices in coding all the time. Thank god Strang's excellent course is available online for free.
"I learned more than enough welding when apprenticing as a metalsmith but now as a chef use none of it."
At least at that time the comp sci people I knew had to take about the same number of math classes the chem people did. Maybe that's changed now.
Re: Too Much Calculus – Gilbert Strang (2001) [pdf]
#95Earlier quoted context omitted.
You are in luck! Strang, Calculus, 2nd edition: http://www.amazon.com/Calculus-Second-Edition-Gilbert-Strang...
And the first edition is freely available at Open Courseware: http://ocw.mit.edu/resources/res-18-001-calculus-online-text...
I wish every day is like this!
Re: Too Much Calculus – Gilbert Strang (2001) [pdf]
#96Earlier quoted context omitted.
It also requires quite a bit of calculus though!
Calculus is indeed a required tool if you want to design new machine learning algorithm. But without linear algebra, good luck implementing an algorithm or even using a library to apply it on a concrete case.
I don't see why this is an either/or decision?
Without a firm understanding of probability then good luck interpreting or fitting a model. Without a firm understanding of linear algebra good luck implementing or using an algorithm.
And hell, if you have a streaming algorithm you probably no longer need linear algebra but to grasp either the probabilistic or geometric interpretations you'll probably need calculus.
Re: Too Much Calculus – Gilbert Strang (2001) [pdf]
#97Earlier quoted context omitted.
I agree although I work in robotics / computer vision. What I wish I had done a lot more of is statistics. Being able to reason about probabilities clearly, getting an idea of how a system performs given various variables, and most importantly, being able to communicate this to others has been very useful to me. I think anyone (front end, backend, etc) can benefit from learning about statistics.
But statistics and probability is really just integration. It's either that or stamp collection. And optimization is really just differentiation (+ some geometry). I'd say the problem is too little calculus and not too much.
Don't know why you're being downvoted. This is totally true.
Re: Too Much Calculus – Gilbert Strang (2001) [pdf]
#98As a programmer, I've found far more uses for linear algebra than for calculus. It shows up in robotics, in machine learning, in game development, and in many other cool subfields. I didn't get very much out of my university linear algebra course. (It was pure theory with nearly zero applications.) I don't think I really developed a good intuition for linear algebra—and why it's so useful—until I read Jim Hefferon's…
As a programmer, I've used calculus and linear algebra exactly never in my professional (20+ years) career. It just doesn't really come up doing web apps, sysadmin or dba work.
I work on games. I use linear algebra daily and calculus frequently, but far less often. Both are extremely relevant to game development although you can get by without calculus and only light linear algebra if you limit your scope or specialize (in something other than graphics or simulation).
Re: Too Much Calculus – Gilbert Strang (2001) [pdf]
#99Earlier quoted context omitted.
I agree although I work in robotics / computer vision. What I wish I had done a lot more of is statistics. Being able to reason about probabilities clearly, getting an idea of how a system performs given various variables, and most importantly, being able to communicate this to others has been very useful to me. I think anyone (front end, backend, etc) can benefit from learning about statistics.
Do you have a recommendation for a good statistics textbook, for someone who is mathematically literate, but doesn't know almost any statistics?
http://smile.amazon.com/Introduction-Bayesian-Statistics-2nd...
Was extremely gentle and mind blowing.
Re: Too Much Calculus – Gilbert Strang (2001) [pdf]
#100I don't think you can cleanly separate the topics. Most applications of linear algebra come from calculus since after all you use calculus to linearize something. In case it was already linear it's probably a linear differential equation -> calculus again. Complex analysis is also closely related to linear algebra via geometric algebra. Rather than teaching them separately and emphasizing linear algebra more, it would make more sense to approach calculus + linear algebra as an integrated subject.