Algebra (from Arabic: الجبر, romanized: al-jabr, lit. 'reunion of broken parts, bonesetting') is one of the broad areas of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. It includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra; the more abstract parts are called abstract algebra or modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics. Abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians.
Elementary algebra differs from arithmetic in the use of abstractions, such as using letters to stand for numbers that are either unknown or allowed to take on many values. For example, in
x
+
2
=
5
{\displaystyle x+2=5}
the letter
x
{\displaystyle x}
is an unknown, but applying additive inverses can reveal its value:
x
=
3
{\displaystyle x=3}
. Algebra gives methods for writing formulas and solving equations that are much clearer and easier than the older method of writing everything out in words.
The word algebra is also used in certain specialized ways. A special kind of mathematical object in abstract algebra is called an "algebra", and the word is used, for example, in the phrases linear algebra and algebraic topology.
A mathematician who does research in algebra is called an algebraist.
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...
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...
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##...
Statement : 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 identity is called but I will take the...
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...
Hello,
While following problem solution found this $$ 4a^4 + 8 a^3 + 8 a^2 + 4a + 1 = ( 2a (a+1) + 1 )^2 $$
Trying to figure out how did author do it but failed.
Anyone?
I find it interesting that so many know how to use all kinds of apps on their cell phones, but so few are able to do simple algebra any more. If you ask around, engineers not included, I think you would find very few people e.g. to be able to find the axis of symmetry of the parabola ##...
What do you guys have to say about this Mathoverflow post?
Do you have any interesting ideas about this?
https://mathoverflow.net/questions/432396/extending-reals-with-logarithm-of-zero-properties-and-reference-request
Please, I have a question about universal enveloping algebra: Let ##U=U(\mathfrak{g})## be the quotient of the free associative algebra ##\mathcal{F}## with generators ##\left\{a_i: i \in I\right\}## by the ideal ##\mathcal{I}## generated by all elements of the form ##a_i a_j-a_j a_i-\sum_{k \in...
Please, in the book of Introduction to Lie Algebras and Representation Theory J. E. Humphreys p.12, I have a question:
Proposition. (3.2). Let ##L## be a Lie algebra.
(c) If ##L## is nilpotent and nonzero, then ##Z(L) \neq 0##.
how we prove this,
Thanks in advance,
Please, in the book of Introduction to Lie Algebras and Representation Theory J. E. Humphreys p.11, I have a question:
Proposition. Let ##L## be a Lie algebra.
(a) If ##L## is solvable, then so are all subalgebras and homomorphic images of ##L##.
(b) If ##I## is a solvable ideal of ##L## such...
Please, in the definition of quotient Lie algebra
If ##I## is an ideal of ##\mathfrak{g}##, then the vector space ##\mathfrak{g} / I## with the bracket defined by:
$$[x+I, y+I]=[x, y]+I, for all x, y \in \mathfrak{g}$$,
is a Lie algebra called the quotient Lie algebra of ##\mathfrak{g}## by...
Please, I am looking for a simple example of derivation on ##sl_2##, if possible, I try to use identity map, but not work with me,
A derivation of the Lie algebra ##\mathfrak{g}## is a linear map ##\delta: \mathfrak{g} \rightarrow \mathfrak{g}## such that ##\delta([x, y])=[\delta(x), y]+[x...
Book answer is ##\qquad a≥0\qquad x={ 7\over 9}\;a\ \ \lor\ \ x={ 13\over 14}\;a\ \ ##but I fail to see how to get there !
Stunned by an 1886 dutch high school exam exercise. Hats off for the 17 year olds that did it !
##\ ##
Please, I need some clarifications about second direction, in the file attached,
$$
\text { Then ad } x \text { ad } y \text { maps } L \rightarrow L \rightarrow I \text {, and }(\text { ad } x \text { ad } y)^2 \text { maps } L \text { into }[I I]=0 \text {. }
$$Thank you in advance,
I found 3 papers on The Exceptional Jordan Algebra in physics
arXiv:2305.00668 (hep-ph)
[Submitted on 1 May 2023]
CKM matrix parameters from an algebra
Aditya Ankur Patel, Tejinder P. Singh
Download PDF
We report a theoretical derivation of the Cabibbo-Kobayashi-Maskawa (CKM) matrix...
The published solutions indicate that the nullspace is a plane in R^n. Why isn't the nullspace an n-1 dimensional space within R^n? For example, if I understand things correctly, the 1x2 matrix [1 2] would have a nullspace represented by any linear combination of the vector (-2,1), which...
For two quantum oscillators, I have raising and lowering operators and , and the number operator . I need to check if operators below follow commutation relations.
Now as far as I know, SU(2) algebra commutation relation is [T_1, T_2] = i ε^ijk T_3. So, should I just get T_1 and T_2 in...
Homework Statement: About semidirect product of Lie algebra
Relevant Equations: ##\mathfrak{s l}_2=## ##\mathbb{K} F \oplus \mathbb{K} H \oplus \mathbb{K} E##
Hi,
Please, I have a question about the module of special lie algebra:
Let ##\mathbb{K}## be a field. Let the Lie algebra...
Homework Statement: please, could you help me to know hoe I compute the Casimir element of lie algebra sl(2), I know the basis and their relations, but i could not find the book explain in details how we get the Casimir element.. I think it is related to killing form, but also I could not find...
I am currently learning some maths from “Precalculus by James Stewart”.
I was wondering if that’s ok? Is it ok to just dive straight into it or go back and brush up my algebra 2 ?
I was wondering what are some good textbooks on algebra 2 by the way?
Thank you.
(This is all for the love of physics).
This is the defining generator of the Lorentz group
which is then divided into subgroups for rotations and boosts
And I then want to find the commutation relation [J_m, J_n] (and [J_m, K_n] ). I'm following this derivation, but am having a hard time to understand all the steps:
especially...
Let ##P(x,y)## be a multivariable polynomial equation given by
$$P(x,y)=52+50x^{2}-20x(1+12y)+8y(31+61y)+(1+2y)(-120+124+488y)=0,$$
which is zero at ##q=\left(-1, -\frac{1}{2}\right)##. That is to say,
$$ P(q)=P\left(-1, -\frac{1}{2}\right)=0.$$
My doubts relie on the multiplicity of this point...
Here is an interesting book a student could do after after Algebra 1, or even integrate into an Algebra 1 course:
https://www.amazon.com/dp/B077VV95N3/?tag=pfamazon01-20
And a website:
https://www.calculussolution.com/
Several topics become easier, such as logarithms, when you know a...
It is easy to see that a matrix of the given form is actually an unitary matrix i,e, satisfying AA^*=I with determinant 1. But, how to see that an unitary matrix can be represented in the given way?
Let ##\Omega## here be ##\Omega=\sqrt{-u}##, in which it is not difficult to realize that ##\Omega ## is real if ##u<0##; imaginary, if ##u>0##. Now, suppose further that ##u=(a-b)^2## with ##a<0## and ##b>0## real numbers. Bearing this in mind, I want to demonstrate that ##\Omega## is real. To...
I'm watching a nice video that tries to explain how linear algebra enters the picture in quantum physics. A quick summary:
Classical physics requires that physical quantities are single-valued and vary smoothly as they evolve in time. So a natural way to model classical physical quantities is...
The toric code is a basic computational model as follows:
There are 2 operations that can be performed, A and B, on this grid.
To compute the value A at each point on the grid, we transform the raw values at each dot (located in between two vertices) according to some predefined operators...
When working with fractions and when we have a fraction or equation with fractions like this one for example ##\frac{x}{x-1}+\frac{x}{x+1}=\frac{9}{4}## do we always need to set boundaries? Like, do we always need to write that x can't be a number that would give the denominator 0? In this...
I came across the following problem somewhere on the web. The original site is long gone.
The problem has me stumped. May be sopmeone can provide some insight.
(The problem seems too simple to post in the "Linear/Abstract Algebra" forum.)
The Cost of Beer
It was nearing Easter, and a group...
Hello,
how can one proof that the dual of ##\mathbb{Z}^n## is ##\mathbb{Z}^n##?
My idea:
The definition of a dual lattice says, that it is as set of all lattice vectors ##x \in span(\Lambda)## such that ##\langle x , y \rangle## is an integer. When we now consider ##\mathbb{Z}^n## we see that...
Well, my guess is that there is something wrong with the factors chosen, because ##\left\Vert \left(0,1,0\right)\right\Vert =1## and
\begin{align}
\left\Vert F\left(0,1,0\right)\right\Vert &=\left\Vert...
Hello,
I am looking for one or more books in combination for self-study of abstract algebra. Desirable would be a good structure of the book with good examples of sentences and definitions. Of course, exercise problems should not be missing.
I am now almost tending to buy the Algebra 0 book by...
I had learned everything in College Algebra in my Algebra II course in high school, and indeed (at least at my alma mater) in engineering, physics or math, no credit is even given for College Algebra.
Perhaps what is going on here is that colleges can't trust that someone who has passed (even...
The following properties of big-O notation follow from the definition:
(i) if ##f(x)=O(u(x))## as ##x\rightarrow{a}##, then ##Cf(x)=O(u(x))## as ##x\rightarrow{a}## for any value of the constant ##C##.
(ii) If ##f(x)=O(u(x))## as ##x\rightarrow{a}## and ##g(x)=O(u(x))## as ##x\rightarrow{a}##...