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

vmls-book.stanford.edu

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

#101

Post by earthicus »

selimthegrim wrote: post #49
earthicus wrote: post #13

Earlier quoted context omitted.

> 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?

Page after page of mathematical insights and delights! I've never had the opportunity to work through it systematically, but have frequently read excerpts and have never been let down. I would expect nothing less from a figure so great as Lax!

It's worth pointing out in the context of this discussion that the book is, by the author's own design, not an introduction to linear algebra. It is a second course that Lax used to teach his advanced undergraduates and beginning graduate students at the Courant Institute. For example, OP with a high school math background will surely be very puzzled by page two, when a linear space is defined as a field 'acting on' a group. Which is, i think, the 'right' way of thinking about the algebraic structure, in the sense that it greatly simplifies all the intricate moving parts of linear algebra. Anyhow, I second your recommendation!

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

#102

Post by lkrych »

muhneesh wrote: post #9
flor1s wrote: post #7

Earlier quoted context omitted.

Don't forget to review calculus as well. Khan Academy is a good start for learning about single variable calculus ( http://www.khanacademy.org ), but their content on multivariable calculus is a bit lacking (neural networks / deep learning use the concept of the derivatives and the gradient a lot). A good supplement for multivariable calculus would be Terence Parr and Jeremy Howard's article on "All the matrix calcul…

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'm going to plug Calculus: Single Variable from the University of Pennsylvania on Coursera (https://www.coursera.org/learn/single-variable-calculus).

This was the best Calculus course I've taken online.

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

#103

Post by edflsafoiewq »

k__ wrote: 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

Linear algebra should be easier than calculus shouldn't it? The whole program of differential calculus is basically that we already know how to solve problems in linear algebra, so let's solve other problems by reducing them to questions of linear algebra in the tangent space.

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

#104

Post by k__ »

edflsafoiewq wrote: post #103
k__ wrote: 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

Linear algebra should be easier than calculus shouldn't it? The whole program of differential calculus is basically that we already know how to solve problems in linear algebra, so let's solve other problems by reducing them to questions of linear algebra in the tangent space.

Sorry, I don't understand.

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

#105

Post by earthicus »

impendia wrote: post #41
k__ wrote: 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…

One difficulty students had that I encountered as a TA was some garbled prerequisites. All of the epsilon-delta definitions are written in propositional logic, and are often '2nd order statements', (that is they have nested quantifiers). This is an entirely new formal language, and its usage is very different than english. It needs to be carefully explained, but standard texts like Stewart just dress it up to look kind of like english and carry on.

In fact the mathematics curriculum DOES acknowledge that you have to teach most students this if you want them to understand it: at my university it lived in the discrete math course, which used Rosen. He devotes an entire chapter on propositional logic, and spends literally 60 pages gradually building up the complexity to arbitrary nested quantifiers; the definition of the limit appears at the end of this.

Unfortunately, discrete math also makes heavy use of sequences and series, so that Calc 2 is a prerequisite of the course... thus my program's student-victims would spend a year taking calculus and not understanding much of the formalism before they were even allowed to take the course that explained the language the basic definitions of calculus were written in! Ugh.

I think Stewart and Rosen are pretty mainstream textbooks, so i suspect this problem is very common. Perhaps you could point it out at your next faculty meeting and shuffle some prerequisites around; we'll start a math-revolution! :)

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

#106

Post by impendia »

dyukqu wrote: post #97
impendia wrote: post #41

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…

> - 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 correc…

I don't know anything about Apostol's Mathematical Analysis. My guess would be that it demands a fairly sophisticated background of the reader, and does an excellent job of covering calculus from an extremely rigorous point of view.

I have heard that Apostol's Calculus is an excellent choice, probably somewhat more accessible to beginners, but still offering a rigorous, highbrow perspective. I've also heard the same of Spivak. I'd probably opt for one or both of these.

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

#107

Post by espeed »

muhneesh wrote: 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:…

Good recommendations. In addition to the UT, MIT and Stanford courses you recommend above, for developing your visual intuition, 3Blue1Brown's Essence of Linear Algebra video series is second to none. [0]

Another good one is MathTheBeautiful [1] by MIT alum Pavel Grinfeld [2]. He approaches Linear Algebra from a geometric perspective as well, but with more emphasis on the mechanics of solving equations. He has a ton of videos organized into several courses, ranging from in-depth Intro to Linear Algebra courses to more advanced courses on PDEs and Tensor Calculus.

Esp note his video on Legendre polynomials [3] and Why {1,x,x²} Is a Terrible Basis: https://www.youtube.com/watch?v=pYoGYQOXqTk&index=14&list=PL....

Gilbert Strang was Greenfield's PhD advisor: https://dspace.mit.edu/handle/1721.1/29345. Pavel has a clear and precise teaching style like Strang, and he makes reference to Prof's Strang and his MIT course from time to time.

NB: Prof Strang has a new book Linear Algebra and Learning from Data that just went to press and will be available in print by mid Jan 2019. A few chapters are available online now, and the video lectures from the new MIT course should on YouTube in a few weeks. [4]

[0] Essence of Linear Video Series (3Blue1Brown): https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw/pla...

[1] MathTheBeautiful https://www.youtube.com/watch?v=pYoGYQOXqTk&index=14&list=PL

[2] https://en.wikipedia.org/wiki/Pavel_Grinfeld

[3] https://en.wikipedia.org/wiki/Legendre_polynomials

[4] MIT Linear Algebra and Learning from Data (2018) http://math.mit.edu/~gs/learningfromdata/

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

#108

Post by joppy »

VikingCoder wrote: post #89
chengiz wrote: post #82

Earlier quoted context omitted.

With due respect, who is "we" and what are you talking about? The book does not as much as mention a homogeneous coordinate and uses 2-arrays for 2D vectors.

I'm off on a tangent unrelated to the book. "we" is computer programmers who deal with vectors and points. Mostly graphics, physics, games, etc.

Technically the projective coordinate (3,2,1) should be exactly the same as (6,4,2), and every nonzero multiple thereof. So it’s not really correct to say that (x, y, 0) represents a vector, or that adding these projective coordinates is vector addition. Vector addition is represented by an identity matrix with the x, y coordinates in the rightmost column.

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

#109

Post by thanatropism »

muhneesh wrote: 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:…

What's the difference between learning and e-learning? Is the latter faster?

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

#110

Post by CamperBob2 »

School-Cotton wrote: post #100
CamperBob2 wrote: post #87

Earlier quoted context omitted.

What is your personal definition of the term "unit vector?" Do you suppose there's a reason why no other textbook or Web site defines it the way these authors do? Are there only N unit N-vectors, as the book says, or are there an infinite number of them?

My definition of unit vectors is the same as yours. However, the book does not say there are only N unit N-vectors. It says there are only N STANDARD unit N-vectors. In math, “Adjective X” usually means something more specific than “X”. “Prime numbers” are a subset of “numbers”, and so on. Just like in this case, “standard unit vectors” are a subset of “unit vectors”.

So why don't they tell readers what unit vectors actually are, in the general case? It's a rather important elementary concept, isn't it?

Why bother describing a specific case using ambiguous language (the parentheses) and omit the one property that actually makes a unit vector a unit vector?

Bad writing in a math textbook is a weird thing to defend.

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