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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
N is a simple nonzero right Amodule (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
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
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
Conjugacy Classes of nonabelian group of order p^3
Do try out this Free Career Guidance Personality Test at https://careertest.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
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
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
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
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 nonzero … Continue reading
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 Algebra, difference of cubes, factorization, video, youtube
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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 Algebra, elementary mathematics, O Level Math Tuition, o level maths tuition
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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
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 A Level Math Tuition, Algebra, Complex number, Education, Math, Mathematics, O Level Math Tuition, Singapore, Student, Teaching Resources
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