Chapter 1 is the foundation upon which everything else will be built

el estudio de extensiones de cuerpos, es decir, en la construcción de nuevos cuerpos que contienen al menos un subcuerpo dado.

Esto sirve para modelar operadores lógicos del álgebra del boole, que son empleados en motores de búsqueda, lógical proposicional, circuitos electrónicos, etc.

Para mayor información sobre el álgebra de Boole ver:

• Operadores booleanos: para qué sirven, lista y ejemplos

• Álgebra de Boole: del silogismo aristotélico a los circuitos integrados

Ken Ribet’s love note from Serge Lang<br /> by Ken Ribet

McCoy, Neal Henry. The Theory of Rings. 1964. Reprint, The Bronx, New York: Chelsea Publishing Company, 1973.

This is a reference to a great book "Beginning Mathematical Logic: A Study Guide [18 Feb 2022]" by Peter Smith on "Teach Yourself Logic A Study Guide (and other Book Notes)". The document itself is called "LogicStudyGuide.pdf".

It focuses on mathematical logic and can be a gateway into understanding Gödel's incompleteness theorems.

I found this some time ago when looking for a way to grasp the difference between first-order and second-order logics. I recall enjoying his style of writing and his commentary on the books he refers to. Both recollections still remain true after rereading some of it.

It both serves as an intro to and recommended reading list for the following: - classical logics - first- & second-order - modal logics - model theory<br /> - non-classical logics - intuitionistic - relevant - free - plural - arithmetic, computability, and incompleteness - set theory (naïve and less naïve) - proof theory - algebras for logic - Boolean - Heyting/pseudo-Boolean - higher-order logics - type theory - homotopy type theory

A vector space is a mathematical structure that allows us to work with things that have both a magnitude and a direction. This is useful for studying physical quantities like forces and velocity. The concept of vector spaces is important for linear algebra, which is a way of solving systems of linear equations.

A vector space is a mathematical structure that allows us to work with things that have both a magnitude and a direction. This is useful for studying physical quantities like forces and velocity. The concept of vector spaces is important for linear algebra, which is a way of solving systems of linear equations.

Notation of a random process that has outcomes

The "universal set" aka "sample space" of all possible outcomes is sometimes denoted by \(U\), \(S\), or \(\Omega\): https://en.wikipedia.org/wiki/Sample_space

Probability theory & measure theory

From what I recall, the notation, \(\Omega\), was mainly used in higher-level grad courses on probability theory. ie, when trying to frame things in probability theory as a special case of measure theory things/ideas/processes. eg, a probability space, \((\cal{F}, \Omega, P)\) where \(\cal{F}\) is a \(\sigma\text{-field}\) aka \(\sigma\text{-algebra}\) and \(P\) is a probability density function on any element of \(\cal{F}\) and \(P(\Omega)=1.\)

Somehow, the definition of a sigma-field captures the notion of what we want out of something that's measurable, but it's unclear to me why so let's see where writing through this takes me.

Working through why a sigma-algebra yields a coherent notion of measureable

A sigma-algebra \(\cal{F}\) on a set \(\Omega\) is defined somewhat close to the definition of a topology \(\tau\) on some space \(X\). They're both collections of sub-collections of the set/space of reference (ie, \(\tau \sub 2^X\) and \(\cal{F} \sub 2^\Omega\)). Also, they're both defined to contain their underlying set/space (ie, \(X \in \tau\) and \(\Omega \in \cal{F}\)).

Additionally, they both contain the empty set but for (maybe) different reasons, definitionally. For a topology, it's simply defined to contain both the whole space and the empty set (ie, \(X \in \tau\) and \(\empty \in \tau\)). In a sigma-algebra's case, it's defined to be closed under complements, so since \(\Omega \in \cal{F}\) the complement must also be in \(\cal{F}\)... but the complement of the universal set \(\Omega\) is the empty set, so \(\empty \in \cal{F}\).

I think this might be where the similarity ends, since a topology need not be closed under complements (but probably has a special property when it is, although I'm not sure what; oh wait, the complement of open is closed in topology, so it'd be clopen! Not sure what this would really entail though 🤷‍♀️). Moreover, a topology is closed under arbitrary unions (which includes uncountable), but a sigma-algebra is closed under countable unions. Hmm... Maybe this restriction to countable unions is what gives a coherent notion of being measurable? I suspect it also has to do with Banach-Tarski paradox. ie, cutting a sphere into 5 pieces and rearranging in a clever way so that you get 2 sphere's that each have the volume of the original sphere; I mean, WTF, if 1 sphere's volume equals the volume of 2 sphere's, then we're definitely not able to measure stuff any more.

And now I'm starting to vaguely recall that this what sigma-fields essentially outlaw/ban from being possible. It's also related to something important in measure theory called a Lebeque measure, although I'm not really sure what that is (something about doing a Riemann integral but picking the partition on the y-axis/codomain instead of on the x-axis/domain, maybe?)

And with that, I think I've got some intuition about how fundamental sigma-algebras are to letting us handle probability and uncertainty.

Back to probability theory

So then events like \(E_1\) and \(E_2\) that are elements of the set of sub-collections, \(\cal{F}\), of the possibility space \(\Omega\). Like, maybe \(\Omega\) is the set of all possible outcomes of rolling 2 dice, but \(E_1\) could be a simple event (ie, just one outcome like rolling a 2) while \(E_2\) could be a compound(?) event (ie, more than one, like rolling an even number). Notably, \(E_1\) & \(E_2\) are NOT elements of the sample space \(\Omega\); they're elements of the powerset of our possibility space (ie, the set of all possible subsets of \(\Omega\) denoted by \(2^\Omega\)). So maybe this explains why the "closed under complements" is needed; if you roll a 2, you should also be able to NOT roll a 2. And the property that a sigma-algebra must "contain the whole space" might be what's needed to give rise to a notion of a complete measure (conjecture about complete measures: everything in the measurable space can be assigned a value where that part of the measurable space does, in fact, represent some constitutive part of the whole).

But what about these "random events"?

Ah, so that's where random variables come into play (and probably why in probability theory they prefer to use \(\Omega\) for the sample space instead of \(X\) like a base space in topology). There's a function, that is, a mapping from outcomes of this "random event" (eg, a role of 2 dice) to a space in which we can associate (ie, assign) a sense of distance (ie, our sigma-algebra). What confuses me is that we see things like "\(P(X=x)\)" which we interpret as "probability that our random variable, \(X\), ends up being some particular outcome \(x\)." But it's also said that \(X\) is a real-valued function, ie, takes some arbitrary elements (eg, events like rolling an even number) and assigns them a real number (ie, some \(x \in \mathbb{R}\)).

Aha! I think I recall the missing link: the notation "\(X=x\)" is really a shorthand for "\(X(\omega)=x\)" where \(\omega \in \cal{F}\). But something that still feels unreconciled is that our probability metric, \(P\), is just taking some real value to another real value... So which one is our sigma-algebra, the inputs of \(P\) or the inputs of \(X\)? 🤔 Hmm... Well, I guess it has the be the set of elements that \(X\) is mapping into \(\mathbb{R}\) since \(X\text{'s}\) input is a small omega \(\omega\) (which is probably an element of big omega \(\Omega\) based on the conventions of small notation being elements of big notation), so \(X\text{'s}\) domain much be the sigma-algrebra?

Let's try to generate a plausible example of this in action... Maybe something with an inequality like "\(X\ge 1\)". Okay, yeah, how about \(X\) is a random variable for the random process of how long it takes a customer to get through a grocery line. So \(X\) is mapping the elements of our sigma-algebra (ie, what customers actually end up experiencing in the real world) into a subset of the reals, namely \([0,\infty)\) because their time in line could be 0 minutes or infinite minutes (geesh, 😬 what a life that would be, huh?). Okay, so then I can ask a question like "What's the probability that \(X\) takes on a value greater than or equal to 1 minute?" which I think translates to "\(P\left(X(\omega)\ge 1\right)\)" which is really attempting to model this whole "random event" of "What's gonna happen to a particular person on average?"

So this makes me wonder... Is this fact that \(X\) can model this "random event" (at all) what people mean when they say something is a stochastic model? That there's a probability distribution it generates which affords us some way of dealing with navigating the uncertainty of the "random event"? If so, then sigma-algebras seem to serve as a kind of gateway and/or foundation into specific cognitive practices (ie, learning to think & reason probabilistically) that affords us a way out of being overwhelmed by our anxiety or fear and can help us reclaim some agency and autonomy in situations with uncertainty.

https://www.loom.com/share/a05f636661cb41628b9cb7061bd749ae

Synopsis: Maggie Delano looks at some of the affordances supplied by Tana (compared to Roam Research) in terms of providing better block-based user interface for note type creation, search, and filtering.

These sorts of tools and programmable note implementations remind me of Beatrice Webb's idea of scientific note taking or using her note cards like a database to sort and search for data to analyze it and create new results and insight.

It would seem that many of these note taking tools like Roam and Tana are using blocks and sub blocks as a means of defining atomic notes or database-like data in a way in which sub-blocks are linked to or "filed underneath" their parent blocks. In reality it would seem that they're still using a broadly defined index card type system as used in the late 1800s/early 1900s to implement a set up that otherwise would be a traditional database in the Microsoft Excel or MySQL sort of fashion, the major difference being that the user interface is cognitively easier to understand for most people.

These allow people to take a form of structured textual notes to which might be attached other smaller data or meta data chunks that can be easily searched, sorted, and filtered to allow for quicker or easier use.

Ostensibly from a mathematical (or set theoretic and even topological) point of view there should be a variety of one-to-one and onto relationships (some might even extend these to "links") between these sorts of notes and database representations such that one should be able to implement their note taking system in Excel or MySQL and do all of these sorts of things.

Cascading Idea Sheets or Cascading Idea Relationships

One might analogize these sorts of note taking interfaces to Cascading Style Sheets (CSS). While there is the perennial question about whether or not CSS is a programming language, if we presume that it is (and it is), then we can apply the same sorts of class, id, and inheritance structures to our notes and their meta data. Thus one could have an incredibly atomic word, phrase, or even number(s) which inherits a set of semantic relationships to those ideas which it sits below. These links and relationships then more clearly define and contextualize them with respect to other similar ideas that may be situated outside of or adjacent to them. Once one has done this then there is a variety of Boolean operations which might be applied to various similar sets and classes of ideas.

If one wanted to go an additional level of abstraction further, then one could apply the ideas of category theory to one's notes to generate new ideas and structures. This may allow using abstractions in one field of academic research to others much further afield.

The user interface then becomes the key differentiator when bringing these ideas to the masses. Developers and designers should be endeavoring to allow the power of complex searches, sorts, and filtering while minimizing the sorts of advanced search queries that an average person would be expected to execute for themselves while also allowing some reasonable flexibility in the sorts of ways that users might (most easily for them) add data and meta data to their ideas.

Jupyter programmable notebooks are of this sort, but do they have the same sort of hierarchical "card" type (or atomic note type) implementation?

https://algebra.org/wp/

https://www.nytimes.com/2001/01/07/education/algebra-project-bob-moses-empowers-students.html

The five step philosophy of the Algebra Project: - physical experience - pictorial representation - people talk (explain it in your own words) - feature talk (put it into proper English) - symbolic representation

"people talk" within the Algebra project is an example of the Feynman technique at work

Link this to Sonke Ahrens' method for improving understanding. Are there research links to this within their work?

Mathematical education programs that are attempting to decrease educational gaps: - Bob Moses' Algebra Project - Adrian Mims' Calculus Project - Jaime Escalante math program - Harlem Children's Zone

https://thinkzone.wlonk.com/Numbers/NumberSets.htm

A relatively clear explanation of the types of numbers and their properties.

I particularly like this diagram:

https://linear.axler.net/LADRvideos.html

I've seen Gilbert Strang's excellent lecture series. I'm curious how these rank in relation.

What does writing an expression using trace operator open up to?

What is the trace operator invariant for?

What is the Frobenius Norm of a Matrix?

The trace operator provides the alternative way of writing which norm of the matrix?

Where the trace operator is useful?

A more intuitive description of why dual spaces are useful or interesting.

一切可能事件的一切组合似乎是个不错的信息的定义。。。

最大的sigma代数是\(\mathcal{F}\)?

• sigma代数的大小如何理解？
• 信息的定义？

$$\left\{\mathcal{I}_{n}, n \geq 0\right\}$$

是一组滤链

• 滤链定义在$$(\Omega, \mathscr{F}, \mathscr{P})$$

• 滤链满足$$\mathcal{I}{0} \subseteq \mathcal{I}{1} \subseteq \mathcal{I}_{2} \subseteq \cdots$$

Does this just look like

[ 1 1 0 0 ]

as in two |0> smooshed together?

Lost me here...

Transpose?

Starting to get a little bit more into linear algebra / complex numbers. I'd like to see this happen more gradually as I haven't used any of this since college.

read while playing with this: http://matrixmultiplication.xyz/

After months of using and learning about matrices, this is the best gist I've come across.

closure in mathematics. sounds similar to domain of a function

I'm curious what year this was, particularly in relation to Claude Shannon's master's thesis in which he applied Boolean algebra to electronics.

Based on their meeting date, it would have to be after 1940. And they published in 1943: https://link.springer.com/article/10.1007%2FBF02478259

李宏毅 linear algebra lec7

Textbook: chapter 1.7

前一节课已经介绍了如何判断【有没有解】：很多“换句话说”

有没有解 ---> 是不是线性组合 ---> 在不在span中。

现在要解决的是：如果有解，那么会有多少个解！

李宏毅 linear algebra lec6: Having solution or Not?

Textbook: chapter 1.6

\(Ax=b\)

能否找到一个 x 使得 \(Ax=b\) 成立.

• Linear combination
• span

有没有解这个问题非常重要：假设 Linear system 是一个电路，现在老板告诉你这个电路要输出 b 这么大的电流，你能不能找到合适的电压源or电流源，还是根本就找不到？

关于“解”的名词定义

consistent

A system of linear equations is called consistent if it has one or more solutions。

只要有解就叫做 consistent.

inconsistent

A system of linear equations is called inconsistent if its solution set is empty(no solution)

没有解就叫做 inconsistent.

如何确定“解”

Naive 方法：线的交点

把 system of linear equations 的方程都画成直线，如果他们有交点，那么就是有解，否则无解。

General 方法

定义引入：Linear Combination

Given a vector set \(\{u_1,u_2,...,u_k\}\)

The linear combination of the vectors in the set: \(v=c_1u_1+c_2u_2+...+c_ku_k,\ c_1,c_2,...,c_k\ are\ scalars\ coefficients\ of\ linear\ combination\)

linear combination is a vector.

有了 Linear combination 的定义之后，我们再回一下 lec5 篇末讲解的关于 使用 column view of product of matrix and vector 所以我们可以得到的结论是：

\(Ax\) 其本质就是一个 linear combination, 他是

• 以 \(x\) 的每一位为 scalar coefficient of linear combination,
• 以 columns of \(A\) as vectors 作为 vector set engaged in linear combination, 的一个 linear combination

矩阵与向量的乘法就是对矩阵的列做线性组合。

对于 \(Ax=b\) 是否有解（x是变量）这件事，实际就是在问：b 是否是columns of A的所有可能的线性组合中的一种。

从是否有解到是否是线性组合

如果两个向量不是平行的同时不是0向量，那么他们可以组合出二维空间中所有可能的向量（亦即，线性组合的所有可能性覆盖整个2D空间）。

【判断题】：如上所说，如果非零非平行的两个向量的线性组合可以覆盖整个二维空间的话，那么非零非平行的三个向量的线性组合是否可以覆盖整个三维空间呢？

【答案】：否

引入 independent 向量

在三维空间中对参与线性组合的向量不能仅仅给出【非零】【非平行】两个限制，还得加上一个【不在同一个二维平面】。试想，如果三个向量处在同一平面的话，那么不论如何线性组合都不可能与第三维有任何关系。

引入 反之不反

非零非平行 ===> 有解；有解 ==X==> 非零非平行。

引入 span

vector set 的所有可能的 linear combination （另一个vector set）就是这组 vector set 的 span。

\(v = c_1u_1+c_2u_2+...+c_ku_k\)

\(v\) 毫无疑问是一个向量。

如果我们穷举所有可能的\(c_1,c_2,...,c_k\)，他们所得到的向量的集合（vector set \(V\)）就是\(x_1,x_2,...,x_k\)的span，同时，\(x_1,x_2,...,x_k\) 叫做 vector set \(V\) 的 generating set.

引入 generating set

\(if\ Vector\ set\ V=Span(S),\ then\ V\ is\ Span\ of\ S, also\ S\ is\ a\ generating\ set\ for\ V,\ or\ S\ generates\ V\)

\(S\) 可以作为一种描述 \(V\) 特性的方法。为什么我们需要这种描述方法呢？因为 \(V\) 作为一个 span，他通常都非常非常的大（一般都是无穷多个），如果我们想要描述这种无穷大（“无穷”都意味着抽象）的向量的集合，最好的方法就是找到一个更具体（“有限”意味着具体）的可联想的“指标” --- generating set --- 这个向量集合是由什么样的向量集合生成的。

相同的向量集(span)可能由不同的向量集（generating set）产生：

\(S_1=\begin{vmatrix} 1 \\ -1\end{vmatrix}\)

同

\(S_2=\{\begin{vmatrix}1\\-1\end{vmatrix},\begin{vmatrix}-2\\2\end{vmatrix}\}\)

产生的向量集是相同的。

引入 span of standard vector

standard vector 其实就是 one-hot encoding vector. 可以见下：

\(e_1=\begin{vmatrix}1\\0\\0\end{vmatrix}, e_1=\begin{vmatrix}0\\1\\0\end{vmatrix}, e_1=\begin{vmatrix}0\\0\\1\end{vmatrix}\)

\(span(e_1)=one\ R^1\ in\ R^3\), one axis in 3D-space \(span(e_1,e_2)=one\ R^2\ in\ R^3\), one 2D-space in 3D-space \(span(e_1,e_2,e_3)=R^3\), whole 3D-space.

其实今天学的东西就是“换句话说”，

• \(Ax=b\) has solution or not?

换句话说

• is \(b\) the linear combination of columns of \(A\)?

换句话说

• is \(b\) in the \(span\) of the columns of \(A\)?

李宏毅 linear algebra lec 5

重新定义线性代数

第三节课讲过，一个线性系统不仅仅是一条“直线”，直线只是一种特殊到不能再特殊的情况。线性系统的本质是：

1. '->' 以下表示线性系统

2. 符合加法性：x->y ==> x1+x2->y1+y2

3. 符合乘法(scalar)性：x->y ==> x1k->yk

广义向量

再结合一个超级牛逼的观点广义向量 --- 函数也是一种向量。我们就把线性系统是一条直线的观点边界向外扩展了一些：

线性系统是以向量（亦即，包含函数和数字和普通向量）作为输入

现实世界中的很多东西都可以表示为向量，就连函数也不例外。

他可以造就这样的奇迹：

1. 加法性：fn->fc ===> fn1 + fn2-> fc1+fc2

2. 乘法性：fn->fc ===> fn1k->fc1k

也就是说，线性系统接收的输入和输出都是一个向量，而数字和函数只是特殊的向量。，满足这一特殊性质的线性系统就是【微分】and【积分】。微分和积分更像是一种【功能】而不是一个【函数】，这也是为什么我们不把系统说成函数的原因，因为他强调功能而不是记号表示性，或者说函数只是功能的一个可记号话的特例

线性代数这门学科研究的主要目标就是线性系统。

于是新的关于线性系统的定义至此形成：

\(vector\ \Rightarrow LinearSystem\ \Rightarrow vector\)

\(domain\ \Rightarrow LinearSystem\ \Rightarrow co-domain\)

线性系统与联立线性等式

可以证明的是（in lec3）任何线性系统都可以表示为联立线性等式，也就是说联立等式与线性系统是等价的

Linear system is equal to System of linear equations.

【矩阵，联立方程式，线性系统】其实是一个东西

1. 矩阵 符合加法/乘法性 所以其为一个线性系统
2. 联立方程式 符合加法/乘法性 所以其为一个线性系统

因为

矩阵=线性系统，

联立方程=线性系统，

所以

矩阵=联立方程。

lec5: 两种方式理解 matrix-vector product

• 可以按行看待matrix,正常看法；
• 可以按列看待matrix,把整个matrix看成一个row向量;

联立方程式 ---> 按列看待matrix的 product of matrix and vector ---> 联立方程式可以写成 Product of matrix and vector. 因为之前说过任何一个线性系统都可以写成联立方程式，那么矩阵就是一个线性系统。

\(Ax=b\) 中的 \(A\) 就是一个线性系统

YES! In American schools you are indoctrinated with the premise: "There is no difficult material. There are only difficult learners." The "trial-by-failure" prevalent in the 70's and 80's, that if you repeat a subject you truly do not nor will not ever understand, Algebra in my case, you are somewhat "subversive" to the rest of classroom, the teacher and especially the school. Report card comments: asks too many questions/asks no questions, disruptive/sleeps in class, no effort given, won't get tutored after school labels the learner without labelling the conformity of the classroom: fit in or be shut out. Excellent point!

2. 綫性相加（combinations），伸展（span）和單位矢量 l 綫性代數的本質 第二章

本节介绍三个相互依存的概念：单位向量，span，线性无关。

基于单位向量和数字的向量的表示

• basis vector \(\hat{i}\)
• basis vector \(\hat{j}\)
• adding together two scaled vectors

是一种新的看待线性代数的观点，非常重要的三个知识点，至此向量的表示可以变成在各个单位向量做放缩然后取和，或者，单位向量的线性组合：

\((-5)\hat{i} + (2)\hat{j}\)

可以表示为：

$$ \begin{vmatrix} -5 \\ 2 \end{vmatrix} $$

what if we choose different basis vectors?

虽然不论使用什么方向的两个单位向量，其线性组合始终可以覆盖全部二维空间，但是我们仍然得到了同一个向量的两个不同的表示：

although \((3.1)\hat{i} + (-2.9)\hat{j} = \(-0.8)\hat{i}+(1.3)\hat{j}\) 但是该向量的实际表示却完全不同：

$$ \begin{vmatrix} -0.8 \\ 1.3 \end{vmatrix} \neq \begin{vmatrix} 3.1 \\ -2.9 \end{vmatrix} $$

所以这里需要给出一种关于线性代数的数字表示法\([3.1, -2.9]\)的一个基本条件：每当使用这种表示法时都必须明确单位向量是什么。

span of vectors

可以想象的是：

• 如果两个单位向量之间存在夹角那么他们的线性组合形成的向量一定可以覆盖整个平面；
• 如果两个单位向量处在同一个方向（相同or相反）那么他们的线性组合形成的向量只能覆盖这条直线
• 如果两个单位向量都是 \(\vec{0}\)，那么他们的线性组合形成的向量都是\(\vec{0}\)

引入概念span

The "span" of \(\vec{v}\) and \(\vec{w}\) is the set of all their linear combinations:

\(a\vec{v} + b\vec{w}\)

let \(a\) and \(b\) vary over all linear numbers.

两个向量的 span 与另一个表述是等价的，仅仅通过加法和乘法两种操作可以产生的所有向量

Vectors VS. Points

【tips】如果仅仅考虑一个向量，经常将向量想象成带箭头线段；如果考虑一堆向量的集合，经常将向量想象成点。

• 那么两个同方向的向量的span就形成一条直线
• 那么两个不同方向的向量的span就形成一个平面
• 那么三个不同方向的向量的span就形成一个体

Redundant and Linearly dependent

任何时候如果你有多个向量，但是去掉其中一个或几个，前者和后者的span没有减少（span is essencially a set --- set of all possible linear combination）

\(span(\vec{v},\vec{w},\vec{u})=span(\vec{v},\vec{w})\)

那么就可以说这个向量与其他向量是 Linear dependent （线性相关）， 或者说这个（可以去掉的）向量可以表示为其他向量的线性组合, 因为这个可以去掉的向量处在其他向量的span中。

\(redundant\ \vec{u} \in span(\vec{v}, \vec{w})\)

或者说，他对扩大span(set of linear combination of vectors)没有作用。

由此衍生出另一个概念：Linearly independent

Linearly independent

\(\vec{u} \neq a\vec{v} + b\vec{w},\ for\ all\ values\ of\ a\ and\ b\)

如果某个单位向量无法通过其他单位向量的任何一种系数的线性组合来得到，那么就说这个向量与其他向量都是线性无关的

basis vector

有了之前的 span linearly dependent 两个概念，下面才能正式定义第三个概念：何为 basis vector ？

The basis of a vector space is a set of linearly independent vectors that span the full space

This is interesting!

A First Course in Linear Algebra, Robert A. Beezer<br> Free as PDF or online.