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The Little Book of Linear Algebra

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Re: The Little Book of Linear Algebra

#61

Post by nh23423fefe »

jcranmer wrote: post #53
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. It's only difficult if you are wedded to a description of matrices and vectors as seas of numbers that you grind your way through without trying to instill a fuller understanding of what those numbers actually mean. The definition makes a lot more sense when you see a matrix a…

I dont agree with this. Matrices don't convert sets of basis vectors to sets of basis vectors. What would you say about singular matrices for example?

The natural motivation of matrices is as representing systems of equations.

Re: The Little Book of Linear Algebra

#62

Post by russellbeattie »

Someone should convert all the examples into C code so it's more intelligible to programmers who are, let's admit, the main audience for something like this.

To the best of my knowledge: Scalars are variables. Vectors are arrays. Matrices are multi dimensional arrays. Addition and multiplication is iteration with operators. Combinations are concatenation. The rest like dot products or norms are just specialized functions.

But it'd be nice to see it all coded up. It wouldn't be as concise, but it'd be readable.

Re: The Little Book of Linear Algebra

#63

Post by JadeNB »

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.

I think that, to be frank, it's a combination of (1) a curriculum developed before it was clear how ubiquitous linear algebra would become, and (2) the fact that it's a lot easier to come up with a standardized assessment for algorithmic calculus than for linear algebra, precisely because linear algebra is both conceptual and proof-based in a way that has been squeezed out of algorithmic calculus.

(I use algorithmic calculus to describe the high-school subject, and distinguish it from what in American universities is usually called "analysis," where one finally has the chance to make the acquaintance of the conceptual and proof-based aspects squeezed out of algorithmic calculus.)

Re: The Little Book of Linear Algebra

#64

Post by JadeNB »

andrewla wrote: post #18
Chinjut wrote: post #15

Earlier quoted context omitted.

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".

Sure, that's a way to approach it. All you have to do is stay interested in "linear functions" long enough to get there. It's totally possible -- I got there, and so did many many many other people (arguably everyone who has applied mathematics to almost any problem has). But when I was learning linear algebra all I could think was "who cares about linear functions? It's the simplest, dumbest kind of function. In fac…

> But when I was learning linear algebra all I could think was "who cares about linear functions? It's the simplest, dumbest kind of function. In fact, in one dimension it's just multiplication -- that's the only linear function and the class of scalar linear functions is completely specified by the factor that you multiple by".

This seems to make it good motivation for an intellectually curious student—"linear functions are the simplest, dumbest kind of function, and yet they still teach us this new and exotic kind of multiplication." That's not how I learned it (I was the kind of obedient student who was interested in a mathematical definition because I was told that I should be), but I can't imagine that I wouldn't have been intrigued by such a presentation!

Re: The Little Book of Linear Algebra

#65

Post by BenFranklin100 »

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…

Probability and statistics falls into that category too. It’s one of the more boring undergraduate math courses, but is mind-boggling useful in the real world.

(Basic probability / combinatorics is actually pretty cool, but both tend to be glossed over.)

Re: The Little Book of Linear Algebra

#66

Post by CamperBob2 »

russellbeattie wrote: post #62

Someone should convert all the examples into C code so it's more intelligible to programmers who are, let's admit, the main audience for something like this. To the best of my knowledge: Scalars are variables. Vectors are arrays. Matrices are multi dimensional arrays. Addition and multiplication is iteration with operators. Combinations are concatenation. The rest like dot products or norms are just specialized funct…

That's basically what you'll get if you pick up a book on 3D game programming. However, progress will come to a halt when you get to things like determinants and eigenvalues that don't show up in core 3D graphics pipelines. You'll have to find other ways to motivate a C version of that part of the curriculum... but I agree, that's a well-worthwhile thing to ask for.

Re: The Little Book of Linear Algebra

#67

Post by CamperBob2 »

relaxing wrote: post #13
rw_panic0_0 wrote: post #7

how is it beginner friendly, first paragraph and already an obscure formula for non math people

I agree. I wouldn’t consider someone who has taken (and remembers) a course in set theory a beginner without some added qualifier. One of my pet peeves is using mathematical symbols beyond basic arithmetic without introducing them once by name. Trying to figure out what a symbol is and what branch of math it comes from is extremely frustrating.

I haven't taken any courses in set theory, but it makes perfect sense to me. Once someone tells you that the funny script 'R' means "Real numbers", the funny E means "is an element of", and the vertical | means "given that," that's pretty much all you need to know to dive in.

If those concepts cause difficulty, it probably makes sense to go back down the learning curve a bit before tackling linear algebra. Alternatively, just cut and paste the expression into any LLM and it'll explain what's what.

Re: The Little Book of Linear Algebra

#70

Post by jcranmer »

nh23423fefe wrote: post #61
jcranmer wrote: post #53

Earlier quoted context omitted.

> Even the "why does matrix multiplication look that way" is incredibly deep but practically impossible to motivate from other considerations. It's only difficult if you are wedded to a description of matrices and vectors as seas of numbers that you grind your way through without trying to instill a fuller understanding of what those numbers actually mean. The definition makes a lot more sense when you see a matrix a…

I dont agree with this. Matrices don't convert sets of basis vectors to sets of basis vectors. What would you say about singular matrices for example? The natural motivation of matrices is as representing systems of equations.

If I write a vector v = [1, 3, 2], what I am actually saying is that v is equal to 1 * e₁ + 3 * e₂ + 2 * e₃ for three vectors I have previously decided on ahead of time that form an orthonormal basis of the corresponding vector space.

If I write a matrix, say, this:

  [[1  2]
   [3  4]
   [5  6]]
What I am doing is describing is a transformation of one vector space into another, by describing how the basis vectors of the first vector space are represented as a linear combination of the basis vectors of the second vector space. Of course, the transformed vectors may not necessarily be a basis of the latter vector space.

> The natural motivation of matrices is as representing systems of equations.

That is very useful for only very few things about matrices, primarily Gaussian elimination and related topics. Matrix multiplication--which is what the original poster was talking about, after all--is something that doesn't make sense if you're only looking at it as a system of equations; you have to understand a matrix as a linear transformation to have it make sense, and that generally means you have to start talking about vector spaces.

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