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Introduction to Applied Linear Algebra: Vectors, Matrices, and Least Squares

vmls-book.stanford.edu

41–50 of 126 posts

Re: Introduction to Applied Linear Algebra: Vectors, Matrices, and Least Squares

#41
post #39

Somehow I found linear algebra easier than calculus, but I don't know why. I did both at the same time in university, but failed calculus 3 times and aced linear algebra at the first try. I'd expect being either good or bad at math, not both at the same time

Math professor here ---

Quality of teaching might have something to do with it.

But, also, calculus is much harder to understand at a rigorous, formal level than at an informal level.

On one level you can try to understand what the main concepts are about, be able to compute derivatives and integrals, solve optimization and related rates problems, and so on. I'd recommend Silvanus Thompson's Calculus Made Easy over any mainstream calculus book for this. In my opinion, the book succeeds amazingly at fulfilling the promise of its title.

But suppose you really try to read any mainstream calculus book, and understand everything. For example:

- Why are limits defined the way they are (with epsilons and deltas)?

- The book will probably touch lightly upon the Mean Value Theorem -- why is this important? What's the point?

- Why is the chain rule true? It reads dy/dx = (dy/du) (du/dx). Yay! This is just cancelling fractions, right? Any "respectable" calculus book will insist that it's not, but most students will cheerfully ignore this, still get correct answers to the homework problems, and sleep fine at night.

- Consider the function e^x. How is it defined? The informal way is to say e = 2.71828... and we define exponents "as usual". Most students are perfectly happy with this. But does this really make sense if x is irrational? Your calculus book might bend over backwards to define everything properly (e^x is the inverse to ln(x), which is defined as a definite integral), and it takes a lot of work to appreciate why.

In my experience, these sorts of issues mostly don't pop up in linear algebra, where the proofs tend to parallel the handwavy heuristics. I wonder if this had anything to do with your experience?

Re: Introduction to Applied Linear Algebra: Vectors, Matrices, and Least Squares

#42
Kind of inauspicious to see this kind of thing on page 5:

A (standard) unit vector is a vector with all elements equal to zero, except one element which is equal to one

Huh? I've never heard the term "standard vector" before, and a "unit vector" is a vector whose magnitude is one. There is no requirement that one element be equal to one.

Maybe that's why the download is free...

Re: Introduction to Applied Linear Algebra: Vectors, Matrices, and Least Squares

#43
> 2-vector (x1,x2) can represent a location or a displacement in 2-D...

Isn’t this fundamentally faulty? Same notation describing a point and displacement. From this, we may conclude that, a point and a displacement are the same thing because they are described by the same notation. Shouldn’t mathematics be free of such contextual interpretation?

Re: Introduction to Applied Linear Algebra: Vectors, Matrices, and Least Squares

#44
post #39

Somehow I found linear algebra easier than calculus, but I don't know why. I did both at the same time in university, but failed calculus 3 times and aced linear algebra at the first try. I'd expect being either good or bad at math, not both at the same time

Math professor here --- Quality of teaching might have something to do with it. But, also, calculus is much harder to understand at a rigorous, formal level than at an informal level. On one level you can try to understand what the main concepts are about, be able to compute derivatives and integrals, solve optimization and related rates problems, and so on. I'd recommend Silvanus Thompson's Calculus Made Easy over a…

I was really bad at rearanging terms/formulars and this came up in calculus exams all the time, but not so much in linear algebra exams.

Re: Introduction to Applied Linear Algebra: Vectors, Matrices, and Least Squares

#45
Applied linear algebra is such a great idea. Linear algebra is relatively easy to understand and used everywhere. But the material is so damn boring since it's a lot of arthimetic. Even the homework problem is boring since there is no specific purpose.

Re: Introduction to Applied Linear Algebra: Vectors, Matrices, and Least Squares

#46
post #9

Earlier quoted context omitted.

Thanks - I am doing that as well! I've been using MIT OpenCourseware for single variable calculus (and will do the same for multivariable). I fenced the parent post to Linear Algebra to not go too far away from the OP. I will certainly check out the Terrence Parr / Jeremy Howard site, and am super familiar with Khan Academy.

I came across-Calculus Made Easy by Silvanus Thompson,on someones twitter feed. Published in 1910 and far less scary and far more interesting to read than a lot of math text books. https://www.gutenberg.org/files/33283/33283-pdf.pdf

There is also a web goodlooking version, discussed previously in HN: https://news.ycombinator.com/item?id=18250034

Re: Introduction to Applied Linear Algebra: Vectors, Matrices, and Least Squares

#47

Kind of inauspicious to see this kind of thing on page 5: A (standard) unit vector is a vector with all elements equal to zero, except one element which is equal to one Huh? I've never heard the term "standard vector" before, and a "unit vector" is a vector whose magnitude is one. There is no requirement that one element be equal to one. Maybe that's why the download is free...

> A (standard) unit vector...

I know this as "an element of the standard basis," B = {e_1, e_2, ...), where e_1 = (1,0,0,...), e_2 = (0,1,0,0,...). You could view it as inauspicious that the treatment doesn't begin with abstract vector spaces, but there is always Axler.

For what it's worth, I find it inauspicious that after taking three (pure-math oriented) Linear Algebra courses I never saw least squares nor the SVD. I'm looking forward to taking a look at Prof Boyd's book.

Re: Introduction to Applied Linear Algebra: Vectors, Matrices, and Least Squares

#48
post #44

Earlier quoted context omitted.

Math professor here --- Quality of teaching might have something to do with it. But, also, calculus is much harder to understand at a rigorous, formal level than at an informal level. On one level you can try to understand what the main concepts are about, be able to compute derivatives and integrals, solve optimization and related rates problems, and so on. I'd recommend Silvanus Thompson's Calculus Made Easy over a…

I was really bad at rearanging terms/formulars and this came up in calculus exams all the time, but not so much in linear algebra exams.

That's another thing I observed teaching calculus: a lot of students who have problems, are having their difficulties with the algebra they supposedly know, and not so much with the "new" material. In theory, we expect students to already be fluent in this sort of algebraic manipulation; in reality, we recognize that a calculus class provides our students an opportunity to improve at this.

I'm not sure I have any suggestion other than lots of practice. And know that you're not alone, this is a completely natural difficulty.

Re: Introduction to Applied Linear Algebra: Vectors, Matrices, and Least Squares

#49
post #4

I'm e-learning Linear Algebra right now to have a good math foundation for Machine Learning. I was a History and Sociology major in college - so I didn't take any math. If you are like me, and working off an initial base of high school math, I would recommend the following (all free): Linear Algebra Foundations to Frontiers (UT Austin) Course: https://www.edx.org/course/linear-algebra-foundations-to-fro... Comments:…

> This course [Strang] is apparently the holy grail course for Intro Linear Algebra. I haven't watched his lectures, but I TA'd a linear algebra course that used his text book, and strongly disliked his presentation. I've heard that's a fairly common reaction actually - it's one of those love it or hate it books. I'm bringing it up because if you (or someone else reading this) turn out to be in the group that doesn't…

Where’s the love for Lax?

Re: Introduction to Applied Linear Algebra: Vectors, Matrices, and Least Squares

#50
post #4

I'm e-learning Linear Algebra right now to have a good math foundation for Machine Learning. I was a History and Sociology major in college - so I didn't take any math. If you are like me, and working off an initial base of high school math, I would recommend the following (all free): Linear Algebra Foundations to Frontiers (UT Austin) Course: https://www.edx.org/course/linear-algebra-foundations-to-fro... Comments:…

If you care about anything that runs on a computer, linear algebra is one of the best maths.
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