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The Little Book of Linear Algebra

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Re: The Little Book of Linear Algebra

#91

Post by ViscountPenguin »

jameshart wrote: post #56
andrewla wrote: post #14

It's crazy that Linear Algebra is one of the deepest and most interesting areas of mathematics, with applications in almost every field of mathematics itself plus having practical applications in almost every quantitative field that uses math. But it is SOOO boring to learn the basic mechanics. There's almost no way to sugar coat it either; you have to learn the basics of vectors and scalars and dot products and matr…

What I find amazing is, given how important linear algebra is to actual practical applications, high school math still goes so deep on calculus at the expense of really covering even basic vectors and matrices. Where vectors do come up it’s usually only Cartesian vectors for mechanics, and only basic addition, scalar multiplication and component decomposition are talked about - even dot products are likely ignored.

That's very strange, where I live linear algebra was a significant portion of the highschool maths curriculum.

The actual presentation was terrible, I'll be lucky if I die before having to invert a matrix by hand again, but it was there.

Re: The Little Book of Linear Algebra

#92

Post by rramadass »

joshmarlow wrote: post #55
andrewla wrote: post #14

It's crazy that Linear Algebra is one of the deepest and most interesting areas of mathematics, with applications in almost every field of mathematics itself plus having practical applications in almost every quantitative field that uses math. But it is SOOO boring to learn the basic mechanics. There's almost no way to sugar coat it either; you have to learn the basics of vectors and scalars and dot products and matr…

The older I get the more convinced I am that "math is not hard; teaching math is hard".

This is far more truer than most people may realize.

Because there is so much to teach/learn, "Modern Mathematics" syllabi has devolved into giving students merely an exposure to all possible mathematical tools in an abstract manner, dis-jointly with no unifying framework, and no motivating examples to explain the need for such mathematics. Most teachers are parrots and have no understanding/insight that they can convey to students and so the system perpetuates itself in a downward spiral.

The way to properly teach/learn mathematics is to follow V.I.Arnold's advice i.e. On Teaching Mathematics - https://dsweb.siam.org/The-Magazine/All-Issues/vi-arnold-on-... Ground all teaching in actual physical phenomena (in the sense of existence with a purpose) and then show the invention/derivation of abstract mathematics to explain such phenomena. Everything is "Applied Mathematics", there is no "Pure Mathematics" which is just another name for "Abstract Mathematics" to generalize methods of application to different and larger classes of problems.

Re: The Little Book of Linear Algebra

#93

Post by zkmon »

In terms of understanding why something is like that, Linear Algebra belongs to Geometry more than Algebra. Every formula in Linear Algebra which ultimately is justified by "it's just that way" can be better justified by geometry. The algebraic formulas are like animals who lost their natural habitat and were put in a zoo, making people think that these animals evolved in the zoo itself.

To give an example: A simple multiplication of two numbers is better seen as rotating one of the numbers to be perpendicular to the other and then quantifying the area/volume spanned by them. This gives vector dot product.

While geometry might better address "why", algebra gets into the work of "how to do it". Mathematics in old times, like other branches of science, did not encourage "why". Instead, most stuff would say "This is how to do it, Now just do it". Algebra probably evolved to answer "how to do it" - the need to equip the field workers with techniques of calculating numbers, instead of answering their "why" questions. In this sense, Geometry is more fundamental providing the roots of concepts and connecting all equations to the real world of spatial dimensions. Physics adds time to this, addressing the change, involving human memory of the past, perceiving the change.

Re: The Little Book of Linear Algebra

#94

Post by 127 »

andrewla wrote: post #14

It's crazy that Linear Algebra is one of the deepest and most interesting areas of mathematics, with applications in almost every field of mathematics itself plus having practical applications in almost every quantitative field that uses math. But it is SOOO boring to learn the basic mechanics. There's almost no way to sugar coat it either; you have to learn the basics of vectors and scalars and dot products and matr…

I had the opposite experience when learning linear algebra as I was also doing 3D computer graphics at the time. It was super interesting and fun. I guess you just have to find an application for it.

Re: The Little Book of Linear Algebra

#96

Post by hiAndrewQuinn »

andrewla wrote: post #14

It's crazy that Linear Algebra is one of the deepest and most interesting areas of mathematics, with applications in almost every field of mathematics itself plus having practical applications in almost every quantitative field that uses math. But it is SOOO boring to learn the basic mechanics. There's almost no way to sugar coat it either; you have to learn the basics of vectors and scalars and dot products and matr…

Linear algebra was such a tedious thing to learn I skipped over it to abstract algebra and doubled back once I had some kind of minimally interesting framework to work with it against. Normally I think this is a foolish way to do things, but sometimes things are just so dull you have to take the hard route to power through at all.

Re: The Little Book of Linear Algebra

#97

Post by dreamcompiler »

andrewla wrote: post #40
srean wrote: post #17

Earlier quoted context omitted.

> Even the "why does matrix multiplication look that way" is incredibly deep but practically impossible to motivate from other considerations. You just start with "well that's the way it is" and grind away In my experience it need not be like that at all. One can start by defining and demonstrating linear transformations. Perhaps from graphics -- translation, rotation, reflection etc. Show the students that these fol…

> Perhaps from graphics -- translation, rotation, reflection Maybe ... but the fact that you included translation in the list of linear operations seems like a big red flag. Translation feels very linear but it is emphatically not [1]. This is not intended to be a personal jab; just that the intuitions of linear algebra are not easy to internalize. Adding linear transformations is similarly scary territory. You can m…

To me the fact that translation in n dimensions is nonlinear but it becomes linear if you embed your system in n+1 dimensions is one of the coolest results from linear algebra, and it's why you need 4x4 matrices to express the full set of transformations possible in 3-space.

Re: The Little Book of Linear Algebra

#98

Post by jonahx »

andrewla wrote: post #14

It's crazy that Linear Algebra is one of the deepest and most interesting areas of mathematics, with applications in almost every field of mathematics itself plus having practical applications in almost every quantitative field that uses math. But it is SOOO boring to learn the basic mechanics. There's almost no way to sugar coat it either; you have to learn the basics of vectors and scalars and dot products and matr…

> Even the "why does matrix multiplication look that way" is incredibly deep but practically impossible to motivate from other considerations.

Short, simple answer to that question by Michael Penn: https://www.youtube.com/watch?v=cc1ivDlZ71U

Another interesting treatment by Math the World: https://www.youtube.com/watch?v=1_2WXH4ar5Q&t=4s

There's no impenetrable mystery here. Probably just bad teaching you experienced.

Re: The Little Book of Linear Algebra

#99

Post by jonahx »

For an unconventional but beautiful presentation of the subject, I highly recommend "Wild Linear Algebra" by the brilliant Norman Wildberger:

https://www.youtube.com/watch?v=yAb12PWrhV0&list=PLBQcPIGljH...

It starts with the axioms of being able to draw one line parallel to another, and a line through point, and builds up everything from there. No labeled Cartesian axes. Just primitive Euclidean objects in an affine space.

Re: The Little Book of Linear Algebra

#100

Post by hinkley »

andrewla wrote: post #14

It's crazy that Linear Algebra is one of the deepest and most interesting areas of mathematics, with applications in almost every field of mathematics itself plus having practical applications in almost every quantitative field that uses math. But it is SOOO boring to learn the basic mechanics. There's almost no way to sugar coat it either; you have to learn the basics of vectors and scalars and dot products and matr…

There was a brief time when I understood how linear algebra was used to render 3d models but that's all gone now.

I wonder if these days that would be a better starting point.

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