Geometric approach for Linear Algebra

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>when the graduate student in charge of the class decides to teach linear algebra as set theory
Holy fuck what a nightmare.

>>8998241
lol turbo autism

>>8998241
That sounds like fun. What is the rigorous set representation of a matrix?

>>8998257
>What is the rigorous set representation of a matrix?
I still don't know. I gave up trying to understand her lectures and watched MIT opencourseware videos instead.

>>8998284
Fuck. My professor just told me that a matrix is an array of elements of a field.

And if you try to google "set representation of a matrix" you will literally find nothing. You should have at least paid attention to that part, given that it is knowledge that is nowhere to be found.

>>8998296
>given that it is knowledge that is nowhere to be found
learn to read a book my man
http://gen.lib.rus.ec/

>>8998315
I've read a couple of linear algebra textbooks. None of them give a set representation for matrices.

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>>8998241

>>8998257
Let [math]X[/math] be a set, then [math]M:\{1,2,...,m\}\times\{1,2,...,n\}\to X[/math] is [math]m\times n [/math] matrix with entries in [math]X[/math]. And obviously function [math]f:X\to Y[/math] is defined as set [math]\{(x, y)|x\in X, y\in Y, for any (x_1, y_1), (x_2,y_2)\,x_1=\x_2\implies y_1=y_2\}[/math]

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>>8998400
me no understand

>>8998406
do you understand the a sequence is a just map [math]\mathbb{N} \to \mathbb{R}[/math] ?

>>8998400
Yeah, this looks familiar. Now imagine trying to take notes on a firehose of that stuff written in tiny faint handwriting on a chalkboard that you can't see because the tiny classroom is overfilled. And then she erases it all so she can write even more. :^)

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>>8998417
me no

>>8998400
That's good, it is clearly a way to express what we intuitively mean by matrix.

The only problem I see with it is that then to construct matrices you first need the natural numbers. And natural numbers are too powerful, too complex. You shouldn't need such a complex construction before you can express matrices.

This can be resolved for things like ordered pairs. For example, you can define [math] (a,b) = \{ \{a\} , \{a, b\}\} [/math]. No need to use natural numbers.

Thus, as your construction needs natural numbers I consider it non satisfactory.

>>8998424
what's wrong with natural numbers? you don't natural number per se, I mean you don't need the fancier stuff like addition, successor function etc. you just need a set containg exactly n elements. and the natural number n, defined set theoretically, is really the easiest construction of such set.

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>>8998284
>MIT opencourseware lin.algebra videos
Wew lad, hope you are not in STEM

>>8998438
I don't like that either because now to define matrices you need a countable set.

Perhaps we could define
[math]\left( \begin{array}{ccc}
a & b \\
c & d \end{array} \right) = \{\{a\}, \{\{a\},\{b\}\},\{ \{ \{a\}, \{b\}, \{c\}\}\}, \{\{\{ \{a\},\{b\},\{c\},\{d\} \}\}\} \}[/math]

and then generalize

>>8998452
1) where in my definition do I need a countable set ? I'm not the guy who suggested the definition by the way, but you should know that it's a standard definition.
2) what's wrong with a countable set ?

>>8998471
>1) where in my definition do I need a countable set ?

Let m and n go to infinity and now you need at least a countable set to keep finding indices.

>2) what's wrong with a countable set ?

Too powerful. Remember the definition of order pairs I shared? Another one is [math] (a,b) = \{ \{a,1 \}, \{b,2\} \} [/math].

This one came before the one I showed before, but it never became widely accepted (unlike the one I first presented) because it needs the natural numbers. And technically you shouldn't need something so complex as numbers to define fundamental mathematical structures.

>>8998443
There's literally nothing wrong with MIT OCW

>>8998484
Of course not, but most video lectures are from babby-tier courses

>>8998482
>Let m and n go to infinity and now you need at least a countable set to keep finding indices.

yeah well of course if you want an infinite matrix you would need at least a countable set. are you telling me you can construct an infinite matrix without a countable set ?

>>8998494
No, I mean to that to keep making bigger matrices you need bigger and bigger numbers. And to not run out of indices, you need at least a countable set.

>>8998501
and how does this not apply to your construction ? you are suggesting some iterative process. all I'm doing is that I'm isolating this process and applying it to my particular problem, because it actually makes the solution nicer and I can use the process later on. I'll try to put it in another words. How do you define a vector or an ordered n-tuple. Of course you can generalize the definition of an ordered pair using nested sets and call this an ordered n-tuple. I would say denote [math]n = \{ \emptyset, \{ \emptyset\},\dots\}[/math] (this is my template nested set) and then I would say that that an ordered n-tuple is a map [math]n \to X[/math]. But you see that we are doing exactly the same thing ? And that I don't need any countable set whatsoever ? The only difference is that your definition is a "shorter code" but my is "more readable".

>And technically you shouldn't need something so complex as numbers to define fundamental mathematical structures.
yes, you shouldn't need the SET of natural numbers, but I don't need this, I just need "numbers" as a "meta definition" for my own convenience, but nothing complex is going on. On the meta level, a fundamental property of a matrix is the NUMBER of its rows and columns so of course you need numbers in SOME way. Without this, the best you can do is to define a [math]X[/math] by [math]Y[/math] matrix with entries in [math]A[/math] as a mapping [math]X \times Y \to A[/math].

>>8998066
I don't know about the book but the videos are dope. I hope he makes a series on algebraic topology one day.

>>8998424
To do anything useful with matrices we want X to be ring, so if we don't think rings are too complex structures then natural numbers are neither

>>8999196
thanks bud. Do you think the videos alone are good enough to learn Linear Algebra?

Anyone else?

>>8998424
i dont quite get your objection to the natural numbers but ignoring that, how would you define an [math]m\times n[/math] matrix using ordered pairs/ x-tuples?
something like a matrix is an m-tuple of n-tuples where each n-tuple is associated with a matrix row? if that makes sense

>>8998424
>You shouldn't need such a complex construction before you can express matrices.
Why not?

>>9000320
you've answered yourself. a matrix is a m-tuple of its rows and a n-tuple of its columns. but as I've already said multiple times, this is unnecessarily messy and it's more convenient to use mappings.

>>8998066
The problem with geometric intuitions is that they're useful except when they aren't. You can't go above 3 dimensions with geometric intuitions.
To be honest, I wish I'd seen that polynomial when I was in high-school because it always confused the fuck out of me how sin(x) and sin^-1 (x) were ACTUALLY calculated.
I think a good linalg course uses geometric intuitions where it's convenient, but the focus should definitely be on the math.

>>9002478
so would you suggest a different book?

>>9002874
Read Lang

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>>8998241
I'M FUCKING SOLD

>>8998452
Can't you define matrices as a set of vectors ?
Seems like it's much easier

>>8998241
>unable to understand the most basic application of set theory
you're fucked and should stop studying math now

>>8998066
any representation of linear algebra which does not take a strict theoretical approach is not worth studying.

t. math grad

>>8998400
>Using "for any" in latex
You had one job, anon.

You're missing part of the definition too. In particular that for every x in X there exists a y such that (x,y) is in f. In other words; f is defined for every element in the domain. In other words, the domain = the pre-image.

>>8998424
>intuitive
I would disagree. It becomes worse when you introduce infinite matrices and so on.

I can somewhat agree with your arguments against natural numbers but since we're in set theory it should be possible to construct them and get rid of that problem.

In general I just feel like this construction in Set Theory is extremely hamfisted (no offense to you or anyone else, of course; I'm aware this construction is considered standard).

>>9002961
Assuming that when you say vectors you mean n-tupples, It's pretty much the same thing.

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>>9003010
that always annoyed me about spivaks definition

>>9003077
Holy shit, that definition is garbage. I've only skimmed Spivak, not actually read it, but from what I saw it's full of shit like this. Doesn't even introduce an axiomatic system for the reals. I don't get why people think that book is great.

>>9003010
>>9003077
this is the definition computer scientists use (at least here). they dick around with their "partial" and "total" functions.

>>9003103
Computer scientists are far more pedantic than mathematicians. You're right that they define partial functions and total functions as special types of partial functions but they would never define anything over "numbers". Depending on the context they will either define functions over arbitrary sets or specifically over the integers (with tupples of integers and other non-integer sets just being encoded as integers in some way).

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>>9003010
It is pretty intuitive, though I think pic related probably makes it clearer

>>9003126
>i for elements in J_m
>j for elements in J_n
>a as a function
>a(i,j) = a_{ij} presented as an equivalence and not just a notational convention

It's not a terrible definition, but it is still hamfisted when it comes to working with them. Unlike number sequences, which are defined as a function from the integers to a set, one never refers to the function directly ever again and never has a need to define the value at a coordinate of a matrix based on the coordinate itself (as one does with number sequences).

>>9002963
can you make any suggestions for textbooks?

>>9003604

linear algebra: "git 'r done" by axler and whitney

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>>9003085
Is this bait, or are you refering to the omission of the completeness axiom?

>>8998241
Is he doing everything over division rings?

>>9004038
It's been a long time since I looked at it so I honestly didn't even remember it did the ordered field axioms (together with the retarded version of the trichotomy law). I assure you that whenever I looked at it I would've been pissed that it omits the completeness axiom since a ton of proofs fall apart without it (you can't even prove basic shit like the archimedean property).

Steven R. Lay's Analysis book is very similar to Spivak but better. Though it still has it's share of flaws.

>>8998319
>>8998296
It's not shown in LA, it's shown in abstract algebra, but on page 2 & 3 of Matrices & Linear Algebra by schneider it talks about it a little.

>>8998257
Ax = b

>>9004333
but he does include the completeness axiom, just in a later chapter

>>8998296
>And if you try to google "set representation of a matrix" you will literally find nothing
>>8998257

It's arbitrary. You could define it as an order NxM tuple over F or as a function f: NxM -> F (the latter eventually generalize to functional analysis)