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Finding the change of basis matrix to convert the matrix form of a linear transformation from one basis to another was hard for me. It still sometimes warps my brain. So I think I will write this down here. Notations and Definitions Consider $V$ an $n$-dimensional vector space over $\mathbb{C}$. Let $\alpha = {a_1, a_1, \ldots, a_n}$ be an ordered basis for our vector space. By the definition of basis, every element $v \in V$ can be uniquely written as a linear combination of basis elements. We define the co-ordinates of $v \in V$ w.r.t basis $\alpha$ to be $[v]_\alpha = (\alpha_1, \alpha_2, \ldots, \alpha_n)^T$ where $\alpha_i$ are scalars such that $v = \sum_{i = 1}^n \alpha_i a_i$. By the uniqueness of representation of vectors as linear combination of basis elements, we get that $A_\alpha: V \to \mathbb{C}^n : v \to [v]_\alpha$ is a linear isomorphism. ...

Things to remember Every C* homomorphism is a bounded linear map which preserve multiplication and involution. Every C* algebra is a Banach algebra, and therefore also a Banach space under the norm. Every * homomorphism from C* algebra to the set of complex numbers is a linear functional of this Banach space. Every * homomorphism between C* algebras are norm decreasing. (Thm 2.1.7, Murphy) Hence every multiplicative * linear functional of this C* algebra is in the closed unit ball of the dual space of the Banach space. A hermitian linear functional are those linear functional (need not be a * homomorphism) which satisfy $f(a) = \overline{f(a^*)}$ for all $a$ in the C* algebra. Positive linear functionals of a C* algebra are those linear functionals (need not be a * homomorphism) where the image of the positive elements of the C* algebra lie the non-negative real line. Clearly positive linear functionals are hermitian. Every multiplicative * linear functional is positive. A bounded linear functional $\phi$ in a unital C* algebra is positive if and only if $|\phi| = \phi(1)$. (Cor. 3.3.4, Murphy) Every hermitian linear functional on a C* algebra can be written as the difference of two positive linear functionals. (Jordan decomposition, Thm 3.3.10, Murphy) Every linear functional $f$ on a C* algebra can be written as $f = g+ih$ where $g, h$ are hermitian linear funcional. States are defined to be those positive linear functionals with norm 1. Convex combinations of states are again states. Pure states of a C* algebra are those states which cannot be written as a convex combination of any other state. So essentially every multiplicative * linear functional on a C* algebra can be written as a linear combination of pure states. ...

These are the steps I followed to root my old Samsung Galaxy J6 device. The procedure might vary slightly for other devices Requirements Stock ROM of the current device OS Android Development Bridge (ADB) installed in computer Heimdall in computer Magisk app installed in device Process Extract boot.img from the current ROM Download the ROM online from SamFW or GalaxyFirmware. It’ll be a .tar.md5 file extract the AP the file using tar -xvf. From the extracted AP get the boot.img.lz4 by the same method lz4 -d boot.img.lz4 to get boot.img Instead of getting the whole ROM, one can just download the AP file to start with. Connect the android device to the computer via ADB Send the boot.img to the Android device. Can use adb push boot.img /sdcard Select Install in the Magisk app and select boot.img and get a patched boot image get the patched boot image back to the computer adb pull /sdcard/Download/Magisk-...img Install heimdall/Odin in computer Put the device to download mode by adb reboot bootloader When the device boots into bootloader mode, verify heimdall detects it by heimdall detect When the device boots into bootloader, flash the boot image using heimdall heimdall flash --BOOT Magisk-...img The android device will auto reboot. If Samsung Knox triggers and ask to factory reset, proceed with it and reinstall Magisk app. Follow the instructions in the Magisk app. Later I also managed to install Magisk in a 2017 J5 Pro following the Magisk wiki ...

A pinch of envy One day I saw my friend Ashish Kujur working with $\LaTeX$ to make notes of the lectures we had. It looked cool. I wanted to learn $\LaTeX$ and make notes like that for a long time. Making proper notes seemed a good way to my goal. I forked his repo. The plan was to pull from his repo, make my edits, add my observations and later push to my repo. That definitely required a computer. ...

What is the serenity that falls with the night? What is the wonder that catches our eye with every twinkle of a star? What is the beauty that draws us again into that good night? Look above; there’s a lion within the stars claiming his part of the sky with a majestic pride. Look again; there’s Orion and his dog, hunting a bull, tales of our ancestors etched in the domes of heaven. And then there’s the inseparable Castor and Pollux, known for their ventures together, set on the sky by the mighty Zeus himself. Like a Shakespearian drama, there are tales of romance, heroism, betrayal, war, bloodshed, and death in the night sky. These stories add a complex layer of perception to the visual beauty of the blinking stars in a dark void. They are reminiscent of the history of humanity and the cultures that prevailed. They remind us of our ancestors’ lives, sharing tales like these around bonfires at night. ...

This is me testing $\LaTeX$ for my blog. Don’t judge me for this. :) Question: If $A_1 =_c A_2$ and $B_1 =_c B_2$ prove that $(A_1 \to B_1 ) =_c (A_2 \to B_2 )$. Or in other words If the cardinality of $A_1$ and $B_1$ are equal to that of $A_2$ and $B_2$ respectively, then the cardinality of the set of functions from $A_1 \to B_1$ is equal to that of the set of functions from $A_2 \to B_2$ (from Notes on Set Theory, Yiannis Moschovakis, Chapter 2, Exercise 2.33) ...