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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

#121

Post by nyc111 »

earthicus wrote: post #119
nyc111 wrote: post #118

Earlier quoted context omitted.

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 ou…

> You represent a vector by giving two pieces of data, (1) an ordered list of coordinates, and (2) a basis.

Ok, I understand. But as used in computer languages, a vector can be simply 2 numbers. No coordinates or basis are implied. That's what I meant.

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

#122

Post by nyc111 »

Koshkin wrote: post #56
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?

My advice would be not to get stuck on these “philosophical” questions, if your goal is to actually learn math, and instead just press on and keep learning and solving real problems. Eventually the fog will dissolve by itself, and these kinds of questions will seem to you either naive or devoid any real substance, or just uninteresting compared to everything else that you have learned.

No. This advice does not apply to me. I don't want to learn mathematics. I'm more interested in learning parts of mathematics that interest me at the moment. And I think philosophy comes before mathematics. Or mathematics is the philosophy of quantities. Both philosopnies are based on definitions.

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

#123

Post by selimthegrim »

graycat wrote: post #120
selimthegrim wrote: post #114

Earlier quoted context omitted.

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.

He passed recently, but yes it’s that one. The book is from 1994 and the research is from just before USSR fell.

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

#124

Post by earthicus »

nyc111 wrote: post #121
earthicus wrote: post #119

Earlier quoted context omitted.

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 ou…

> You represent a vector by giving two pieces of data, (1) an ordered list of coordinates, and (2) a basis. Ok, I understand. But as used in computer languages, a vector can be simply 2 numbers. No coordinates or basis are implied. That's what I meant.

Aha, yes. Computer languages borrowed the word 'vector', but they have basically nothing to do with the mathematical structure from linear algebra. It's best to keep them completely separate in your mind.

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

#125

Post by nyc111 »

earthicus wrote: post #124
nyc111 wrote: post #121

Earlier quoted context omitted.

> You represent a vector by giving two pieces of data, (1) an ordered list of coordinates, and (2) a basis. Ok, I understand. But as used in computer languages, a vector can be simply 2 numbers. No coordinates or basis are implied. That's what I meant.

Aha, yes. Computer languages borrowed the word 'vector', but they have basically nothing to do with the mathematical structure from linear algebra. It's best to keep them completely separate in your mind.

So in math, when we say "vector" coordinate system is a given, as you explained?

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

#126

Post by earthicus »

nyc111 wrote: post #125
earthicus wrote: post #124

Earlier quoted context omitted.

Aha, yes. Computer languages borrowed the word 'vector', but they have basically nothing to do with the mathematical structure from linear algebra. It's best to keep them completely separate in your mind.

So in math, when we say "vector" coordinate system is a given, as you explained?

If a coordinates are given, then they will be given with respect to a basis. However, it's entirely possible to do things more abstractly without introducing coordinates and bases to begin with, for example:

https://en.wikipedia.org/wiki/Tensor_(intrinsic_definition)

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