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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

#111

Post by selimthegrim »

graycat wrote: post #77
Koshkin wrote: post #60

Earlier quoted context omitted.

> determinants... CAN be omitted Also see http://www.axler.net/DwD.html .

Looks like a nice paper! The paper says how to go beyond what is in Boyd, et al. , i.e., eigenvalues, eigenvectors, the spectral decomposition, etc. without determinants. Nice! For that material I would have been tempted just to use the old approach of determinants and the roots of the characteristic polynomial , the Hamilton-Cayley theoem, etc. Saved the paper! Thx.

And if you really really love determinants there's

https://www.amazon.com/Discriminants-Resultants-Multidimensi...

(warning: not for undergraduates)

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

#112

Post by CBLT »

thanatropism wrote: post #109
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?

Orders of magnitude cheaper.

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

#113

Post by graycat »

selimthegrim wrote: post #111
graycat wrote: post #77

Earlier quoted context omitted.

Looks like a nice paper! The paper says how to go beyond what is in Boyd, et al. , i.e., eigenvalues, eigenvectors, the spectral decomposition, etc. without determinants. Nice! For that material I would have been tempted just to use the old approach of determinants and the roots of the characteristic polynomial , the Hamilton-Cayley theoem, etc. Saved the paper! Thx.

And if you really really love determinants there's https://www.amazon.com/Discriminants-Resultants-Multidimensi... (warning: not for undergraduates)

There is also the chapter in P. Halmos, Finite Dimensional Vector Spaces on multi-linear algebra which at the time I read it I took it as an abstract approach to determinants, maybe also a start on exterior algebra of differential forms, but maybe there's a long shot chance that that Halmos chapter is related to multi-dimensional determinants.

Can't read ALL the books on the shelves of the research libraries or even all the recent ones so have to be selective, to focus or as a startup entrepreneur before spending hundreds of hours in such a book (hope the author got tenure) ask "Why should I?".

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

#114

Post by selimthegrim »

graycat wrote: post #113
selimthegrim wrote: post #111

Earlier quoted context omitted.

And if you really really love determinants there's https://www.amazon.com/Discriminants-Resultants-Multidimensi... (warning: not for undergraduates)

There is also the chapter in P. Halmos, Finite Dimensional Vector Spaces on multi-linear algebra which at the time I read it I took it as an abstract approach to determinants, maybe also a start on exterior algebra of differential forms, but maybe there's a long shot chance that that Halmos chapter is related to multi-dimensional determinants. Can't read ALL the books on the shelves of the research libraries or even…

I am sure Gelfand, Kapranov and Zelevinsky given their other math accomplishments all got tenure track positions when they emigrated. Will give Halmos another look.

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

#115

Post by profosaur »

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…

Calculus by Spivak is good. Abbot's real analysis textbook is also quite popular.

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

#116

Post by profosaur »

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.

Great comment, this big picture strategy of calculus is not emphasized enough.

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

#117

Post by profosaur »

k__ wrote: post #104
edflsafoiewq wrote: post #103

Earlier quoted context omitted.

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.

The fundamental strategy of calculus is to replace a nonlinear function with a tangent line approximation to that function. This greatly simplifies calculations, and the approximation is often accurate enough to be useful.

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

#118

Post by nyc111 »

earthicus wrote: post #59
nyc111 wrote: post #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?

There's no issue with the notation; I think you've misunderstood the mathematical idea. Consider a more familiar algebraic object, a real number, x. This can model a length, an area, volume, time, time interval, temperature, weight, speed, physical constant, geometric ratio, fractional dimension, etc... In mathematics, we abstract by forgetting about what the things are , and retain information about how they behave…

Thanks for the explanation. Makes sense. Something else I noticed, vector notation does not specify a coordinate system. V = (1, 2) is just an array of two numbers. The cartesian coordinate interpretation is a choice we make. Correct?

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

#119

Post by earthicus »

nyc111 wrote: post #118
earthicus wrote: post #59

Earlier quoted context omitted.

There's no issue with the notation; I think you've misunderstood the mathematical idea. Consider a more familiar algebraic object, a real number, x. This can model a length, an area, volume, time, time interval, temperature, weight, speed, physical constant, geometric ratio, fractional dimension, etc... In mathematics, we abstract by forgetting about what the things are , and retain information about how they behave…

Thanks for the explanation. Makes sense. Something else I noticed, vector notation does not specify a coordinate system. V = (1, 2) is just an array of two numbers. The cartesian coordinate interpretation is a choice we make. Correct?

Yes, the keyword here is 'basis'. You represent a vector by giving two pieces of data, (1) an ordered list of coordinates, and (2) a basis. The vector is then a linear combination of the basis elements, and the coordinates tell you how to form that linear combination.

For example, let's use the standard Cartesian basis consisting of unit vectors e1, e2, e3 (which point north, east, and up, informally speaking). If our vector v is given by the coordinates (3,4,8) (with respect to the standard basis), then this means that v = 3 * e1 + 4 * e2 + 8 * e3.

If the coordinates were given with respect to a different set of basis vectors, then you would take the linear combination using those vectors instead. Note the similarity of how a basis works to how a base system works representing numbers. Using base 10, the 'coordinates' of the number 348 mean that 348 = 3 * 100 + 4 * 10 + 8 * 1. Using a different base, say base 9, they would instead mean 348 = 3 * 81 + 4 * 9 + 8 * 1.

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

#120

Post by graycat »

selimthegrim wrote: post #114
graycat wrote: post #113

Earlier quoted context omitted.

There is also the chapter in P. Halmos, Finite Dimensional Vector Spaces on multi-linear algebra which at the time I read it I took it as an abstract approach to determinants, maybe also a start on exterior algebra of differential forms, but maybe there's a long shot chance that that Halmos chapter is related to multi-dimensional determinants. Can't read ALL the books on the shelves of the research libraries or even…

I am sure Gelfand, Kapranov and Zelevinsky given their other math accomplishments all got tenure track positions when they emigrated. Will give Halmos another look.

That can't still be THE Gel'fand, along with Kolmogorov, prof of E. Dynkin? Must be a great grand son or some such.
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