updated posts

Author: localhost433

.DS_Store

@@ -1,2 +1,2 @@
-# Blog
-Repository for blog.
+# ~
+Repository for a personal site.

README.md

@@ -18,7 +18,7 @@
     <header>
         <div class="header-title">~</div>
         <div class="header-row">
-            <p>Welcome! This is where I share random ideas and stuff I created about school, life, etc.</p>
+            <p>Welcome! This is where I share random ideas and stuff I created about school, life, etc. (Not any type of category that I want to fall into, journal, diary, because it would feel kind of restrictive on 'how'/'what' should I write about anything.)</p>
             <button id="theme-toggle">Dark Mode</button>
         </div>
     </header>

index.html

@@ -1,7 +1,7 @@
 document.addEventListener("DOMContentLoaded", () => {
   const container = document.getElementById("posts-list");
 
-  fetch("metadata/posts.json")
+  fetch("metadata/entries.json")
     .then((res) => res.json())
     .then((posts) => {
       posts.forEach((post) => {

js/main.js

@@ -2,7 +2,29 @@ window.MathJax = {
     tex: {
         packages: { '[+]': ['ams'] },
         macros: {
-            qed: '\\square'
+            qed: '\\square',
+            ran: '\\text{ran}',
+            N: '\\mathbb{N}', // Natural Numbers
+            Z: '\\mathbb{Z}', // Integers
+            Q: '\\mathbb{Q}', // Rational Numbers
+            R: '\\mathbb{R}', // Real Numbers
+            C: '\\mathbb{C}', // Complex Numbers
+            Li: '\\mathcal{L}', // Linear maps
+            M: '\\mathcal{M}', // Matrix Representation
+            im: '\\text{im }', // Image, Range, Column space
+            rank: '\\text{rank}', // Rank
+            Span: '\\text{span}', // Span
+            sgn: '\\text{sgn}', // sgn function
+            inr: '\\left( #1, #2 \\right)', // Inner Product
+            coord: '\\left( #1, #2 \\right)', // Coordinates
+            norm: '\\left\\lVert #1 \\right\\rVert', // Norm
+            conj: '\\overline{#1}', // Conjugate
+            abs: '\\left| #1 \\right|', // Absolute Value
+            proj: '\\text{proj}_{#1}', // Projection
+            tr: '\\text{Tr}(#1)', // Trace
+            id: '\\text{id}', // Identity
+            bi: '\\left( #1, #2 \\right)', // Bilinear form
+            sig: '\\left( #1, #2, #3 \\right)' // Signature
         },
         inlineMath: [['$', '$'], ['\\(', '\\)']],
         displayMath: [['$$', '$$'], ['\\[', '\\]']]

js/mathjax.js

@@ -28,7 +28,7 @@ document.addEventListener("DOMContentLoaded", () => {
         return;
     }
 
-    fetch(`./posts/${slug}.md`)
+    fetch(`./posts/entries/${slug}.md`)
         .then((res) => res.text())
         .then((markdown) => {
             const { metadata, body: markdownBody } = parseFrontMatter(markdown);

js/posts.js

@@ -1,11 +1,11 @@
 ---
 title: First Post
 date: 2025-03-30
-tags: [cs, blog, life]
+tags: [cs, life]
 author: R
 ---
 
-This is (going to be) the first post, the first thing I'm writing in Markdown. I thought about doing the whole thing in $\LaTeX$, but it would be weird, so yeah... I came to this idea of a blog website only after going down a rabbit hole on reddit, and I start wondering "it would actually be cool if I have some type of project that I work on", so here I am. I felt like I'm only going to learn the practical stuff that I'm actually going to use while doing "side quests" like this that I'm interested in. (Its pretty unlikely I'll have the chance to write an entire linked list structure from the basics after CSCI-UA 102...)
+This is (going to be) the first post, the first thing I'm writing in Markdown. I thought about doing the whole thing in $\LaTeX$, but it would be weird, so yeah... I came to this idea of a blog website only after going down a rabbit hole on reddit, and I start wondering "it would actually be cool if I have some type of project that I work on", so here I am. I felt like I'm only going to learn the practical stuff that I'm actually going to use while doing "side quests" like this that I'm interested in.
 
 (One Random thought, no matter what I write, it appears more formal then I would do while I'm talking.)
 

blog/comments/first.json posts/comments/first.jsonblog/comments/first.json renamed to posts/comments/first.json

@@ -0,0 +1,43 @@
+---
+title: Random stuff in linear algebra
+date: 2025-04-02
+tags: [math]
+author: R
+---
+
+While reading the Wikipedia article about metric spaces just now, it was talking about the motivation like "We can measure the distance between two such points by the length of the shortest path along the surface, "as the COW flies". Then I was like there's something wrong about this, and its 'As the crow flies'. This was kind of random but yeah, I want to learn what is a measure space suddenly because I encountered this thing called a bilinear form.
+
+These are not really on the topic, what I want to say today is about the essential part of a linear transformation, and I figured out the proof and purpose just after looking over it in terms of domains and codomains. What I understand here is that
+
+$$Ax = A(P_{\ran A^*}x + (x - P_{\ran A^*}x)) = AP_{\ran A^*}x$$
+
+just like doing an orthogonal decomposition on $x$, that 
+
+$$x - Proj_{\ran A^*}x \in (\ran A^*)^\perp = \text{ker} A$$
+
+so that
+$A(x - Proj_{\ran A^*}x) = 0$, thus it restricts the vectors in the transformation by removing the ones in $\text{ker} A$, and changing $A$ to $\tilde{A}$ just further restricts the domain because now there's only input from $\ran A^*$. 
+
+The dual space $A'$ ($\Li (V, F)$ space of all the linear functionals) sometimes have the notation $A^*$ which coincides with the adjoint operator, this is what makes me think about the relation between the two. Specifically, the Riesz representation theorem states that for all linear functionals $\phi$ on $V$, there exists a unique $v \in V$ such that:
+
+$$\phi(u) = \inr{u}{v}$$
+
+Assume $A': V \to V^*$ is the dual operator, then
+
+$$(A'\phi)(u) = \phi(A(u))$$
+
+By Riesz,
+
+$$\phi(A(u)) = \inr{A(u)}{v}$$
+
+Then by the property of the adjoint,
+
+$$\inr{A(u)}{v} = \inr{u}{A^*(v)}$$
+
+So,
+
+$$(A'\phi)(u) = \inr{u}{A^*(v)}$$
+
+$\qed{}$
+
+which is quite fascinating to discover on my own.