Algebra Definition and 999 Threads

The screenshots below are taken from the 2nd editon of the book How to Prove it A structured approach By: Daniel Velleman and and 3rd edition of the book's solution manual. (Page 5) The question on page 4 exercise 5b corresponds to the solution in page 5 exercise 6b Page 1 Page 2 Page 3...

Hi, I'm trying to derive the equation (14) ##a^r = \frac{a u^r e}{\sqrt{e^2 + g_{tt}}}## from this article No Way Back: Maximizing survival time below the Schwarzschild event horizon and my algebra is really messy, so I'm wondering if I made some mistakes. The authors say: "With the above...

Grassmman variables are defined as anti-commuting, such that for two Grassmann variables ##\theta_1,\theta_2## such that ##\theta_1\theta_2=-\theta_2\theta_1##. Then for a Grassmann variable ##\theta##, the derivatives are such that $$\frac{\partial }{\partial \theta}\theta=1$$ as in usual...

İf ##-1+\dfrac{2}{1-x}=\sqrt 3## then find ##\dfrac{2x}{1-x^2}\cdot \sqrt 3## I can solve it with two ways: 1) directly by finding ##x## 2) Let ##x=\tan\alpha## and go on and answer is ##1.## But I need another way. Are there an another way?

a-1 x a+1= a squared-1 Is this an established theorem or have I stumbled onto something that might get my name in the math books?

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Not sure where the right place would be to put this, but I just noticed that Borcherds has videos of lectures he has given. I had seen the ones on schemes, but he has now a good selection of topics. They are not only algebraic, so may be the general forum would have been better, but anyway here...

This is from the front cover of Mathematical Conundrums, CRC Press, 2023. << When Holly was twice the age Ivy was 15 years ago, Ivy was half the age Holly will be in 12 years time. How many years ago was that? >> OK, but why isn't the age of Holly and Ivy given?

I have tried ##p(x)=ax^3+bx^2+cx+d## but this is a long way. How can I solve easily that question?

Mentor note: Moved from technical math section, so missing the homework template filling in the boxes is easy, 1,3,6,10,15 second is just squares of that, 1,9,36,100,225 But how I anyone supposed to find and expression for this? This is from a textbook on elementary algebra, the specific...

This is basic modular arithmetic but I just can't get it to work no mather how many different methods I try. I probably have failed to understand some basics of modular algebra... Help is appreciated! Correct is supposed to be ##16##

This is exercise 1.8.3 from Foster & Nightingale: Show that if ##\sigma_{ab} = \sigma_{ba}## and ##\tau^{ab} =-\tau^{ba}## for all ##a##, ##b##, then ##\sigma_{ab}\tau^{ab}=0##. I began writing down ##\sigma_{ab}\tau^{ab}=\sigma_{ba}(-\tau^{ba})=-\sigma_{ba}\tau^{ba}##. Here I got stuck and...

Is there something beyond Holors? :cool:

I solved this exercise a long time ago, and I am now reviewing my solution. I think it is correct, but more importantly, when I look at this exercise now, I have a hard time understanding it and my solution. Hence I've got some questions. If this is correct, why does "if ##a=b##, then ##b=a##"...

Proof: Let ## x=\sqrt[3]{18+\sqrt{325}}+\sqrt[3]{18-\sqrt{325}} ##. Then ## x^3=(\sqrt[3]{18+\sqrt{325}}+\sqrt[3]{18-\sqrt{325}})^3 ##. Note that ## (a+b)^3=a^3+3ab(a+b)+b^3 ## where ## a=\sqrt[3]{18+\sqrt{325}} ## and ## b=\sqrt[3]{18-\sqrt{325}} ##. This gives ##...

I am trying to create a complete college curriculum on abstract algebra in Urdu as there are very few advanced resources for students who want to learn mathematics but are not proficient in English. In South Asia and the Middle East, only privileged people are proficient in English and...

$$(b-a)^2-4(b-c)(c-a)=0$$ $$b^2-2ab+a^2=4(bc-ab-c^2+ac)$$ $$b^2-2ab+a^2+4ab=4bc-4c^2+4ac$$ $$(b+a)^2-4ac=4c(b-c)$$ $$b-c=\frac{(b+a)^2-4ac}{4c}$$ I don't know how to continue and not even sure what I did is useful. Thanks

I've been struggling to understand what was the key insight or insights that linear algebra brought to math, or what problems it allowed the solving of that couldn't be solved before. To make a comparison with calculus, I understand that calculus' two key insights were finding a method to...

hb=54 2h+2b=33 h=54/b therefore, 2(54/b)+b=33 108/b + b = 33 I’ve got a feeling I’ve gone down a blind alley here. Any hints?

My name is Feliz. I am doing a self-study of college algebra. I am not a classroom student. My student days ended in December 1993. I've always enjoyed mathematics and decided to conduct a review of material learned long ago.

With the algebra so(3) are associated the spherical harmonics. I would assume that comparably with the algebra so(2,1) are associated functions that can be addressed as hyperbolic harmonics. But I nowhere found any reference to them. Do they exist and if so, where can they be found? Thank you...

Statement : I copy and paste the problem as it appeared in the text. Attempt : I confess I couldn't go much far at all. Here's my attempt below in ##\text{Autodesk Sketchbook}^{\circledR}##. The underlined , wavy underlined and box brackets below are my attempts to see what terms can be...

I honestly do not understand this question, my thoughts; ignoring the diagram and using algebra i can see that the step size [1,5] → [2,6] can be found by adding 1 (common difference) to each number meaning that the answer is A... ...the other options B,C,D and E can not be related by a...

##\Rightarrow \begin{cases} (\sqrt{\dfrac{x}{y}}+\sqrt{\dfrac{y}{x}})^2=(4\dfrac{1}{4})^2\\ (\dfrac{x}{\sqrt{y}}+\dfrac{y}{\sqrt{x}})^2=(16\dfrac{1}{4})^2 \end{cases}## ##\Leftrightarrow \begin{cases} \dfrac{x}{y}+\dfrac{y}{x}+2=\dfrac{289}{16}\\ \dfrac{x^2}{y}+\dfrac{y^2}{x}...

Question 1: Find the modulus and argument of ##z=-\sin \frac {\pi}{8}-i\cos \frac {\pi}{8}##. The modulus is obviously 1. I can't prove that the argument is ##\frac {-5\pi} {8}##. I think ##\frac {-5\pi} {8}## is not correct ... What I've done: $$\tan \theta=\cot \frac {\pi}{8}$$$$\tan...

I would like to take the derivative of the following function with respect to Gt: $$\mathrm{G}_{t+1}=\mathrm{g}_{0}\mathrm{e}^{-qHt}$$ I think that the answer is either -1 or ##\mathrm{e}^{-qHt}-1## If you could show the calculations that would be a great help. Thanks very much.

I'm trying to brush up on my algebra, trig, and calculus, and one thing I know I was always weak on before was proofs. I was never sure what equations would suffice as "proof," and which equations did not. Maybe this is an inane question, and maybe there is a really simple answer to this. I...

I have a very basic confusion that supports some basic elements of algebra. Being a high school student my teacher couldn't answer this, hope someone could help here. We know this equation is true: (-x)^2=x^2 but once we square root both sides it becomes this: -x=x we can see this equation was...

Statement of the problem : Let me copy and paste to the right the problem as it appears in the text. Attempt : I couldn't go far into the solution. Below is my hopeless attempt. Request : Any hints would be welcome. [There are no solutions provided, but the answers are at the back of the...

First I tried to solve this with algebra, but there are not enough equations: a+ b + c + d + e + f + g + h = 36 S = 12 + (d +f + a)/3 ........... ( d +f + a has to be a multiple of 3) a + b + c = e + f a + h + g = d + e So I had to resort to the trial and error to find the solution...

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If I reason this as follows, I run into problems. Please help me understand what is wrong with reasoning like this. a) I start with the left hand side of the equation and let that x be -2. b) I square it. This gives me 4. So I now have the square root of 4. c) The square root of 4 is +/- 2. The...

Let ## \varphi \subseteq A \times B; \psi \subseteq B \times C ##. Then ## \varphi \circ \psi = \left \{ (a, c)| \exists b: (a,b) \in \varphi, (b,c) \in \psi \right \} \subseteq A \times C##. Task: Let ##\varphi## and ##\psi## are subalgebras of algebras ##A \times B## and ##B \times C##...

[FONT=times new roman]Statement : [FONT=times new roman]Let me copy and paste the statement as it appears in the text on the right. Attempt : I could attempt nothing to prove the identity. The best I could do was to verify it for a given value of the ##a's, m, n##. I am not even sure what this...

It would be nice if someone could find the history of why we use the letters i and j or m and n for the basics when working with Matrices ( A = [aij]mxn ). I tried looking up the information and I was not successful. I understand what they represent in the context of the matter, but not why they...

##x+y:x-y=\dfrac{x+y}{x-y} \tag1## Subtract ##y## from each term: ##x:x-2y=\dfrac{x}{x-2y} \tag2## Assume ##k=\dfrac{x}{y} \Rightarrow x=ky## ##(1)= \dfrac{ky+y}{ky-y}, (2)= \dfrac{ky}{ky-2y}## Subtract ##(1)## from ##(2)## since we are told by the problem statement ##(2)## is bigger...

Is it possible to trace the trajectory of an object using only its velocity and position, both of which are given as components. My method of doing so involves using the time until max height is reached, and using that time value to calculate the max height itself (h,k), then plugging in the...

((x+1)^2 -1)/(x+1)^2 dx 1-1/(x+1)^2 dx Let u=x+1 1-1/u^2 du u+1/u +c (u^2+1)/u +c Not as answer given in the book.

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part (a) ##(4+3x)^{1.5} = 2^3+ 9x+ \left[\dfrac {1}{2} ⋅ \dfrac {3}{2} ⋅\dfrac {1}{2}⋅\dfrac {1}{2}⋅9x^2\right]+ ...## ##(4+3x)^{1.5}=8+9x+\dfrac {27}{16} x^2+...##part (b) ##x≠-\dfrac {4}{3}##part (c) ##(8+9x+\dfrac {27}{16} x^2+...)(1+ax)^2 = \dfrac{107}{16} x^2## ... ##8a^2+18a+\dfrac...

Hello, While following problem solution found this $$ 4a^4 + 8 a^3 + 8 a^2 + 4a + 1 = ( 2a (a+1) + 1 )^2 $$ Trying to figure out how did author do it but failed. Anyone?

I find it interesting that so many know how to use all kinds of apps on their cell phones, but so few are able to do simple algebra any more. If you ask around, engineers not included, I think you would find very few people e.g. to be able to find the axis of symmetry of the parabola ##...

What do you guys have to say about this Mathoverflow post? Do you have any interesting ideas about this? https://mathoverflow.net/questions/432396/extending-reals-with-logarithm-of-zero-properties-and-reference-request

https://upload.wikimedia.org/wikipedia/commons/d/d1/Taxonomy_of_Complex_Matrices.svg

Please, I have a question about universal enveloping algebra: Let ##U=U(\mathfrak{g})## be the quotient of the free associative algebra ##\mathcal{F}## with generators ##\left\{a_i: i \in I\right\}## by the ideal ##\mathcal{I}## generated by all elements of the form ##a_i a_j-a_j a_i-\sum_{k \in...

Please, in the book of Introduction to Lie Algebras and Representation Theory J. E. Humphreys p.12, I have a question: Proposition. (3.2). Let ##L## be a Lie algebra. (c) If ##L## is nilpotent and nonzero, then ##Z(L) \neq 0##. how we prove this, Thanks in advance,

Please, in the book of Introduction to Lie Algebras and Representation Theory J. E. Humphreys p.11, I have a question: Proposition. Let ##L## be a Lie algebra. (a) If ##L## is solvable, then so are all subalgebras and homomorphic images of ##L##. (b) If ##I## is a solvable ideal of ##L## such...

Please, in the definition of quotient Lie algebra If ##I## is an ideal of ##\mathfrak{g}##, then the vector space ##\mathfrak{g} / I## with the bracket defined by: $$[x+I, y+I]=[x, y]+I, for all x, y \in \mathfrak{g}$$, is a Lie algebra called the quotient Lie algebra of ##\mathfrak{g}## by...

Please, I am looking for a simple example of derivation on ##sl_2##, if possible, I try to use identity map, but not work with me, A derivation of the Lie algebra ##\mathfrak{g}## is a linear map ##\delta: \mathfrak{g} \rightarrow \mathfrak{g}## such that ##\delta([x, y])=[\delta(x), y]+[x...

Book answer is ##\qquad a≥0\qquad x={ 7\over 9}\;a\ \ \lor\ \ x={ 13\over 14}\;a\ \ ##but I fail to see how to get there ! Stunned by an 1886 dutch high school exam exercise. Hats off for the 17 year olds that did it ! ##\ ##