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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....

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....

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....

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....

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....

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....