Tag Archives: Algebra

Schur’s Lemma

Schur’s Lemma is a useful theorem in algebra that is surprisingly easy to prove. Schur’s Lemma: Let and be right -modules and let be a nonzero -module homomorphism. (i) If is simple, then is injective. (ii) If is simple, then is … Continue reading

Posted in math | Tagged | Leave a comment

N is a simple nonzero right A-module (Equivalent Conditions)

Let be a nonzero right -module. Then the following are equivalent: (i) is simple. (ii) for all (iii) for some maximal right ideal of . Proof: (i)=>(ii) Let . is a submodule of . (Let and , then , ). … Continue reading

Posted in math | Tagged | Leave a comment

Opposite Algebra

Let be an -algebra. Define an operation by . This algebra is called the opposite algebra to . We verify that it forms an -algebra. Bilinearity comes from the following computations: Associativity is true from The unity element is the … Continue reading

Posted in math | Tagged | Leave a comment

Center of Matrix Algebra / Matrix Ring is Scalar Matrices

We will prove that the center , where is an -algebra (or ring with unity). One direction is pretty clear. Let . Then for some . , , so is in the center . The other direction will require the … Continue reading

Posted in math | Tagged | Leave a comment

Conjugacy Classes of non-abelian group of order p^3

Do try out this Free Career Guidance Personality Test at https://career-test.com/s/sgamb?reid=210 while it is still available! Benefits of doing the (Free) Career Test: Get familiar with the top companies in Singapore There are seven distinct career types based on career preferences, goals … Continue reading

Posted in class equation | Tagged , , | Leave a comment

S_n has Trivial Center for n greater than 3

We will prove that for , . Proof: Clearly, . Let . There exists distinct elements of such that . Consider the transposition , where is distinct from . (Since , such a exists.) Then, . Therefore, Do check out … Continue reading

Posted in permutation | Tagged | Leave a comment

Singapore Haze & Subgroup of Smallest Prime Index

Recently, the Singapore Haze is getting quite bad, crossing the 200 PSI Mark on several occasions. Do consider purchasing a Air Purifier, or some N95 Masks, as the haze problem is probably staying for at least a month. Personally, I … Continue reading

Posted in Algebra | Tagged , , | 4 Comments

Proof of Wilson’s Theorem using Sylow’s Theorem

Wilson’s theorem is a useful theorem in Number Theory, and may be proved in several different ways. One of the interesting proofs is to prove it using Sylow’s Third Theorem. Let , the symmetric group on p elements, where p … Continue reading

Posted in sylow | Tagged , | Leave a comment

Quadratic Equations Questions and Answers

Quadratic Equations Questions and Solutions Today we will discuss about the quadratic equation . The way to derive the Quadratic Formula is to use completing the square. First, we divide throughout by a, where a is assumed to be non-zero … Continue reading

Posted in Algebra, completing the square, e maths tuition, quadratic, quadratic equation | Tagged , , , | Leave a comment

Difference of Cubes Video

A geometric interpretation of the Difference of Two Cubes formula.

Posted in maths tuition, maths tutor, O Level Maths Tuition | Tagged , , , , | Leave a comment

Make (y) the subject of the formulae (O Level Maths Tuition)

Question: Make (y) the subject of the formulae (1) Solution: Cross multiply,

Posted in Algebra | Tagged , , , | Leave a comment

积少成多: How can doing at least one Maths question per day help you improve! (Maths Tuition Revision Strategy)

We all know the saying “an apple a day keeps the doctor away“. Many essential activities, like eating, exercising, sleeping, needs to be done on a daily basis. Mathematics is no different! Here is a surprising fact of how much … Continue reading

Posted in Algebra | Tagged , , , , , , , , , , , , , | Leave a comment

How to avoid Careless Mistakes for Maths?

Many parents have feedback to me that their child often makes careless mistakes in Maths, at all levels, from Primary, Secondary, to JC Level. I truly empathize with them, as it often leads to marks being lost unnecessarily. Not to mention, it … Continue reading

Posted in a level math, Algebra, o level maths | Tagged , , , , , , , , , | 7 Comments