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The Little Book of Linear Algebra

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Re: The Little Book of Linear Algebra

#101

Post by hinkley »

zkmon wrote: post #93

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…

I remember how betrayed I felt when I got to calculus and realized all of those equations I learned in high school for physics were just introductory calculus. Like why did we have to learn all that the hard way?

Re: The Little Book of Linear Algebra

#102

Post by AadiBugg »

cosmic_cheese wrote: post #35
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…

If anybody is aware of materials that teach linear algebra via graphics as suggested here, I would be interested to hear about them. As someone who learns best through practical application, maths have been by far among my greatest weak points, despite having written software for upwards of a decade. It’s limiting in some scenarios and pure imposter syndrome fuel.

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Re: The Little Book of Linear Algebra

#103

Post by srean »

hinkley wrote: post #100
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.

That's just the sad truth of the maximum -- use it or lose it. It can be quite disheartening. The good part is that it becomes easier to pick it up next time around. Happened to me so many times.

Re: The Little Book of Linear Algebra

#106

Post by imtringued »

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…

Actually, the calculation of matrix multiplication is extremely easy to understand if you show it done with a row and column vector first. Once your students understand dot products, they will recognize matrix multiplication as the mere pairwise calculation of dot products.

Re: The Little Book of Linear Algebra

#107

Post by rramadass »

anthk wrote: post #69

Klong/k looks ideal for this to have some practice with linear alg concepts: http://t3x.org/klong/index.html

This was one of the reasons for me wanting to study the J programming language. I have a notion AI/ML programming might be better done using array languages.

Some books for studying Mathematics using J are listed here - https://code.jsoftware.com/wiki/Books

Re: The Little Book of Linear Algebra

#108

Post by imtringued »

srean wrote: post #17
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. 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…

>The beautiful reveal is that this addition and composition of linear transformations behave almost the same as addition and multiplication of real numbers.

This is only beautiful if you already understand monoids, magmas and abelian half groups (semigroups) and how they form groups. Also, we do not talk of linear transformations, we talk of group homomorphisms.

I don't know about anyone else, but I was taught linear algebra this way in the first semester and it felt like stumbling in a dark room and then having the lights turned on in the last week as if that was going to be payback for all the toe stubbing.

Re: The Little Book of Linear Algebra

#109

Post by griffzhowl »

mugamuga wrote: post #95

Is there a version or a similar book that deals with Calculus?

Serge Lang's "Short Calculus" is clear and concise, efficiently covering the basics in ~170 pages instead of hundreds of pages like some books. It then has a few more chapters going into a bit more depth but on topics that are also essential if you want to take things further
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