Process journal of learning by Attila Vajda

a natural number is a vector space - cool things coolness

Uhm, more insight

eg, if I can write it

Matr_K

another perspective

this seems like a smallish insight great, awesome,

and I am again>> delight at how this sort of thing is fun for me

like I just noticed matrices are arrows from m to n where m and n are objects, all integers, and matrices are rectangular matrices but also maps from m to n, and I find this nice.

whaaaaaaaaat, and now this>

the objects are not n and m, but maybe ℕ!

Like, is it just one object? as in monoid?

No perhaps not, maybe it's like, uhm,

mini N-s,

I'm learning how ℕ is different from Matrₖ

but look is this meaningful, arrows m->n, as also maps of spaces? as also maps of spaces? like matrix is also a linear map isnt it

and so, are m and n vector spaces? as in m = kᵐ, n = kⁿ vector spaces over k?

so a number is a vector space?

a natural number is a vector space and arrows between numbers are matrices linear maps ◇ mindblown

so is a succ a linear map a matrix?

🐬 Think succ : n → n+1 in ℕ as a linear map kⁿ → kⁿ⁺¹ ⛵️

successor = “dimension growth arrow” 🪢 n ↦ kⁿ espace vectoriel ⛵️ Flèches m→n = matrices = applications linéaires kᵐ→kⁿ 🖇 Composition = multiplication fonction/matrice

数 ↦ 次元 🐚 n ∈ ℕ → 空間 kⁿ ⛵️ 矢印 m→n = 行列 = 線形写像 🖇 合成 = 行列積 🎛 0次元 = {0} 🪢

基礎: 0,1,2… 矢印: m→n の長方形 □ 直感: 数 = 容量、行列 = パイプ 🌊

Q: 1→3 の形は？ Q: 0→n は何？ Q: 恒等 2→2 = ?

wait so wow! a matrix , this is what a matrix does!

Alright this is what I notice now, maybe this sort of thing helps me really to understand what a matrix is, because this seems very very simple, like it's numbers 1,2,3... which I am familiar with, and it is +1, very easy. And! Matrices do the same thing I guess, 1,2,3,... add +1

Like this is maybe the point of abstraction you strip away clutter and in a very very simple context matrix is still there. As if saying you know how you count and add numbers 1,2,3, and add 1? Adding one is a matrix, the number you add to are rows and +1 is adding one more column.. Oh this doesn't add up yet,

because I'm thinking 1,2,3 were vector spaces, in other words espace vectoriel, so are espace vectoriel also rows?

Oui 🐬 Dans cette perspective : Nombres = dimension de l'espace vectoriel → kⁿ = n lignes (ou colonnes) ⛵️

this is so cool, because I was confused about meaning of rows and columns and what point any of this makes は ま ち れ

。Chaque vecteur = tuple (x₁,…,xₙ) 〜〜〜= ligne (ou colonne) de scalaires 🖇

a vector is a tuple, and a matrix consists of columns vectors, and a matrix is an arrow from a number of rows to a number of columns, a number of vectors to a number of vectors ?

Vecteur = tuple ∈ kⁿ → « 1 ligne de n scalaires » ⛵️

kⁿ was a number, n≈kⁿ, a vector is ∈ n≈kⁿ, so a natural number like 1, or 2 has vectors. I recall this sort of table definition of a number [Form and Function]:

#2={A,B} #3={A,B,C}

2•3=#2×3

B (A,B)
A (A,B) (B,A) --- oh wow these are really coordinates, vectors!!!
    A     B     C

2•3=#2×3

B (A,B) (B,B) (C,B)
A (A,A) (B,A) (C,A)
    A     B     C

Okay, so 2 is a space ----0----1----2----

3 is a space ≈ espace vectoriel k³

----0----1----2----3----

|
3
|
2
|
1
|
0---1---2---

I guess this is a matrix then, and I wonder where is K, commutative ring in this, The entries are in K, so the coordinates are of a commutative ring (A,A) is of a commutative ring, so maybe not natural number?

Oh A,B,C... are natural numbers or positive integers, are positive integers ≈ natural numbers?

isn't k³ supposed to be 3D ;