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The Little Book of Linear Algebra

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Re: The Little Book of Linear Algebra

#21

Post by dapper_bison17 »

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…

> This "little book" seems to take a fairly standard approach, defining all the boring stuff and leading to Gaussian elimination. The other approach I've seen is to try to lead into it by talking about multi-linear functions and then deriving the notion of bases and matrices at the end. Or trying to start from an application like rotation or Markov chains.

Which books or “non-standard” resources would you recommend then, that do a better job?

Re: The Little Book of Linear Algebra

#22

Post by Sharlin »

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…

For anyone who’s interested in graphics programming and/or is a visual learner/thinker, there’s an incredibly motivating and rewarding way to learn the basics of linear algebra. (And affine algebra, which tends to be handwaved away, unfortunately. I’m writing a MSc thesis about this and related topics.)

Re: The Little Book of Linear Algebra

#23

Post by andrewla »

dapper_bison17 wrote: post #21
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…

> This "little book" seems to take a fairly standard approach, defining all the boring stuff and leading to Gaussian elimination. The other approach I've seen is to try to lead into it by talking about multi-linear functions and then deriving the notion of bases and matrices at the end. Or trying to start from an application like rotation or Markov chains. Which books or “non-standard” resources would you recommend t…

I have yet to encounter an approach that is not boring. You just have to power through it. This approach seems as good as any.

Once you get to eigenvalues (in my opinion) things start to pick up in terms of seeing that linear spaces are actually interesting.

This approach sort of betrays itself when the very first section about scalars has this line:

> Vectors are often written vertically in column form, which emphasizes their role in matrix multiplication:

This is a big "what?" moment because we don't know why we should care about anything in that sentence. Just call it a convention and later on we can see its utility.

Re: The Little Book of Linear Algebra

#24

Post by andrewla »

Sharlin wrote: post #22
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…

For anyone who’s interested in graphics programming and/or is a visual learner/thinker, there’s an incredibly motivating and rewarding way to learn the basics of linear algebra. (And affine algebra, which tends to be handwaved away, unfortunately. I’m writing a MSc thesis about this and related topics.)

To a degree I think this is true, but it requires (at least in my experience) that you have an intrinsic grasp of trigonometry for it to make sense. If you have some complex function analysis and e^itheta then you can skirt the problem for a bit, but if you're like me and have to break out soh-cah-toa whenever you break down a triangle then this method ends up being pretty tedious too.

Re: The Little Book of Linear Algebra

#25

Post by [deleted] »

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…

[deleted]

Re: The Little Book of Linear Algebra

#26

Post by cassepipe »

Sharlin wrote: post #22
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…

For anyone who’s interested in graphics programming and/or is a visual learner/thinker, there’s an incredibly motivating and rewarding way to learn the basics of linear algebra. (And affine algebra, which tends to be handwaved away, unfortunately. I’m writing a MSc thesis about this and related topics.)

...

What is this incredible motivating way ? Please do tell

Re: The Little Book of Linear Algebra

#27

Post by cassepipe »

Chinjut wrote: post #15
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…

Why do you say it's practically impossible to motivate matrix multiplication? The motivation is that this represents composition of linear functions, exactly as you follow up by mentioning. It's a disservice to anyone to tell them "Well, that's the way it is" instead of telling them from the start "Look, these represent linear functions. And look, this is how they compose".

Agree.The fact that it's just linear functions is what made it click for me

Re: The Little Book of Linear Algebra

#28

Post by GabriDaFirenze »

barrenko wrote: post #8

Tried to pick a book to get into linear algebra recently, the experience was fairly hellish. First course this, second course that, done right, done wrong... I'd to the LADR4e route, but I don't have the proof-it chops yet...

I've found the "No bullshit Guide to Linear Algebra" pretty good. Could be worth checking it out. It's the one resource that has things click more for me.

Re: The Little Book of Linear Algebra

#29

Post by tptacek »

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 didn't think any part of linear algebra was boring. I was hooked from the moment I saw Ax=b => x = b/A. Gaussian elimination is a blast, like an actually-productive Sudoku puzzle, and once you have it down you can blaze through the first 2/3rds of an undergrad linear algebra course. I don't consciously try to gain automaticity with math subjects, but matrix-column multiplication I got pretty quickly and now I just have it.

I learned from Strang, for what it's worth, which is basically LU, spaces, QR, then spectral.

I am really bad at math, for what it's worth; this is just the one advanced math subject that intuitively clicked for me.

Re: The Little Book of Linear Algebra

#30

Post by winwang »

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 hated precalc, but I loved proof-based linear algebra, my first university math course (linalg + multivariate calc).

The simplicity(/beauty) of matrix multiplication still irks me though, in the sense of "wow, seriously? when you work it out, it really looks that simple?"

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