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.
Apologies if this should be in homework section but I thought it best suited here. Been revising past papers but with no solutions. the following questions all require just a true or false answer. Any help or confirmation of my answers would be appreciated.
1 - every N x N matrix has N...
I know De-Morgan's law that $$ -(p∧q) = -p∨-q $$
Also $$ -(p∨q) = -p∧-q $$
But for material implication and bi conditional operations there are also some transformation.
What is the law or proof for it? Like
$$ p⇒q = -p∨q $$
$$ p ↔q = (p∧q) ∨ (-p∧-q) $$
There may be other properties also that I...
Homework Statement
##J=r^{2}\dot{\phi}## [1]
##\dot{r^{2}}=E^{2}-1-\frac{J^{2}}{r^{2}}+\frac{2MJ^{2}}{r^{3}}+\frac{2M}{r}##. [2]
(the context is geodesic equation GR, but I'm pretty sure this is irrelevant).
where ##u=r^{-1}##
Question: From these two equations to derive...
Homework Statement
[/B]
Matrix A =
1 1
0 1
Matrix B =
a b
c d
Find coordinates on a, b, c, d such that AB = BA.
Homework EquationsThe Attempt at a Solution
I calculated AB and BA with simple matrix multiplication, but am not sure where to go from here.
AB =
a + c a + b...
Hello everybody,
in Schwartz' QFT book it says (p. 483 - 484)
In Problem 25.3 this is repeated asking the reader for a proof. I wonder though if this is really true. I know this can be proven for Lie algebras of compact Lie groups (or to be precise, every representation is equivalent to a...
I found out I can pick up a second major in math should I elect to take a two semester sequence in abstract algebra. My first major is in chemical engineering. Right now, I plan on taking a two semester sequence in either: 1) probability with measure theory, 2) abstract algebra (Dummit and...
Let $P$ be a function defined on $[0, 1]$ such that $P(0)=P(1)=1$ and $|P(a)-P(b)|<|a-b|$, for all $a\ne b$ in the interval $[0, 1]$.
Prove that $|P(a)-P(b)|<\dfrac{1}{2}$.
Given series:1,2,5,12,25...
How did they get :##T_n=a(n-1)(n-2)(n-3)+b(n-1)(n-2)+c(n-1)+d##
And for series like 3,7,13,21,...
they have given ##T_n=an^2+bn+c##
How do you get these equations?
So I have the metric as ##ds^{2}=-(1-\frac{2m}{r})dt^{2}+(1-\frac{2m}{r})^{-1}dr^{2}+r^{2}d\Omega^{2}##*
I have transformed to coordinate system ##u,r,\phi, \theta ##, where ##u=t-r*##(2),
where ##r*=r+2m In(\frac{r}{2m}-1)##
and to the coordinate system ##v,r,\phi, \theta ##,
where...
Homework Statement
I have the expression [((1-x)/x) * x^(2/1-x)] / x^(2x/1-x)
I want to simplify this expression.Homework Equations
None
The Attempt at a Solution
I am not so good with algebra. But I tried to find a common factor for x^(2/1-x)] / (x^2x)/1-x and then add them. However, it...
https://www.quantamagazine.org/20150312-mathematicians-chase-moonshines-shadow/
Sorry for the long title but ST = string theory.
Just thought it was interesting news personally since string theory has been elusively hard to prove or observe(at least the particles it claims to predict, notably...
I would like to know some general properties of the modulo (remainder) function that I can use to rewrite expressions. For example, say we wanted to prove the following by rewriting the right-hand-side:
$$ \Big{\lfloor} \frac{n}{d} \Big{\rfloor} = \frac{n - n \pmod d}{d} $$
I have no idea how...
Hi I got stuck at the following circuit problem which involves linear algebra, since I am not a physics major, I don't even have the basics to get started. Please shed some light, really appreciate it! Thanks a lot.
1. Homework Statement
Consider a long chain of resistors wired up like this...
Hi
It's just that last step I'm not getting, so you have:
[1 / Kz] - [1 / (2K)z]
= [ (2K)z - Kz ] / [(2K)z * Kz]
= [ (2)z - 1 ] / [(2K)z*]
Then what?
Thanks
Let N be a normal subgroup of a group G and let f:G→H be a homomorphism of groups such that the restriction of f to N is an isomorphism N≅H. Prove that G≅N×K, where K is the kernel of f.
I'm having trouble defining a function to prove this. Could anyone give me a start on this?
My book says in the slow motion approx, so ## v << c ##, ##v=\frac{dx^{i}}{dt}=O(\epsilon) ##
It then states:
i) ##\frac{dx^{i}}{ds}=\frac{dt}{ds}\frac{dx^{i}}{dt}=O(\epsilon) ##
ii) ## \frac{dx^{0}}{ds}=\frac{dt}{ds}=1+O(\epsilon) ##
The geodesic equation reduces from...
This problem is so simple that I'm not exactly sure what they want you to do:
Let A and B be n x n matrices such that AB = BA. Show that (A + B)^2 = A^2 + 2AB + B^2. Conclude that (I + A)^2 = I + 2A + A^2.
We don't need to list properties or anything, just manipulate. This all seems...
I am trying to learn the formalism of qm, so i am following the book linear algebra done right but is it worth it to study every proof? I mean what is the attitude to follow with such a proof oriented book to eventually have a solid basis in the libear algebra of qm?
Homework Statement
Homework Equations
A=LU, U^-1 * L^-1= A^-1 , U^-1 * L^-1 * U^-1 * L^-1 = A^-2,
The Attempt at a Solution
I used MATLAB and the relations:
U^-1 * L^-1= A^-1 , U^-1 * L^-1 * U^-1 * L^-1 = A^-2,
to find a solution
I found U^-1*L^-1 , let =B...
I decide to self-study linear algebra. I have heard words about some good books on this subject such as Sheldon Axler's. Unfortunately his book is only loanable for 4 days in my university library. There is this book from S. Lang that I can borrow for one month, so what do you think about this...
Homework Statement
Show that the members of the Lie algebra of SO(n) are anti-symmetric nxn matrices. To start, assume that the nxn orthogonal matrix R which is an element of SO(n) depends on a single parameter t. Then differentiate the expression:
R.RT= I
with respect to the parameter t...
Homework Statement
Show that the given matrices are row equivalent and find a sequence of elementary row ops that will convert A into B.
a =
2 0 -1
1 1 0
-1 1 1
b =
3 1 -1
3 5 1
2 2 0
Homework EquationsThe...
Homework Statement
"Let L1 be the line having parametric equations : x = 2 - s, y = -1 + 2s, z = 1+s and L2 be the line:
x = 1 +t, y = 2+ t, z =2t .
a. Do the lines intersect? If so, find the point of intersection.
b. Find the point P on the graph of L1 that is closest to the graph of L2...
The question word for word :
"Write the equation satisfied by all of the points P(x, y, z) that are at the same distance from the point F(0, 0, 4) and the plane z = 0."
I figured I could maybe start by finding the distance between point F and the plane z = 0 but I can't figure out how to...
Homework Statement
How do you know if say [(x_1,y_1),(x_2,y_2)] = x_1x_2 + 7y_1y_2 ? or any other equation?
Homework EquationsThe Attempt at a Solution
I was given the following as a proof that the inertial tensor was symmetric. I won't write the tensor itself but I will write the form of it below in the proof. I am confused about the steps taken in the proof. It involves grade projections.
A \cdot (x \wedge (x \cdot B)) = \langle Ax(x...
Recently I have known an algebraic theorem on polynomial while learning the method of partial fraction.
The theorem is :
Any polynomial can be written as the product of linear factors and irreducible quadratic factors.
I did not find an intuitive proof of this theorem.I asked a question in this...
Warning: This is going to be a bit long.
(Apparently my post was too long so it wouldn't render at all. I've split this into two threads.)
I worked out some basic Algebraic properties of a Lie Algebra. This is similar to my previous thread about SU(2) but as I don't know this example I'm...
First, let me say that I am a senior physics undergrad. I have failed Linear Algebra once before. Otherwise I am a straight A student. I'm also taking Ordinary Differential Equations right now, and I breeze through that class without a care in the world. I'm not sure if I've developed some sort...
I was taught that when you have a polynomial fraction where the denominator is of a higher degree than the numerator, it can't be reduced any further. This seems wrong to me for a couple of reasons.
1. If the denominator can be factored some of the terms may cancel out
2. Say you have the...
Homework Statement
Prove that if A, B, and C are square matrices and ABC = I, then B is invertible and B−1 = CA.
Homework EquationsThe Attempt at a Solution
I think I have this figured out, just checking it. Heres what I got:
ABC=I
(ABC)B-1=IB-1
(B*B-1)AC=IB-1
I*AC=IB-1 Cancel I using left...
I'm looking at: http://arxiv.org/pdf/gr-qc/9712019.pdf,
deriving the FRW metric, and I don't fully understand how the Ricci Vectors eq 8.5 can be attained from 7.16, by setting ##\partial_{0} \beta ## and ##\alpha=0##
I see that any christoffel symbol with a ##0## vanish and so so do any...
Homework Statement
(a) Show acceleration is perpendicular to velocity
(b)Show the following relations
(c) Show the continuity equation
(d) Show if P = 0 geodesics obey:
Homework EquationsThe Attempt at a SolutionPart (a)
U_{\mu}A^{\mu} = U_{\mu}U^v \left[ \partial_v U^{\mu} +...
Homework Statement
I am confused a little bit.
I have been asked to simplify this function: A/B + /BC + /AB I seriously can't seem to know how to simplify this as for me its already simplified.
Homework Equations
N/A
The Attempt at a Solution
The best I could do is : (/A+/B) * (A+B+C) I did...
Homework Statement
Two pumps of different sizes can empty an entire fuel tank in 4.8 hours. Used alone, the larger pump would empty the tank 4 hours less than would the smaller pump. If using only the smallest pump How long will it take to empty the tank?
Homework Equations
X = small pump...
Homework Statement
Prove that set of all onto mappings of A->A is closed under composition of mappings:
Homework Equations
Definition of onto and closure on sets.
The Attempt at a Solution
Say, ##f## and ##g## are onto mappings from A to A.
Now, say I have a set S(A) = {all onto mappings of A...
Homework Statement
Prove that if A is an n x n matrix with the property A3=A, then det(A)=-1, det(A)=0, or det(A)=1
Homework EquationsThe Attempt at a Solution
At first I started with the property A3=A
I then applied the determinant to both sides.
From this point I don't really see any...
<< Mentor Note -- OP has been advised to type their questions into the forum next time, instead of inserting images >>
1. Homework Statement
Dear Mentors and PF helpers,
I have a question from today's lesson.
Homework EquationsThe Attempt at a Solution
Is 1) and 2) both acceptable? If...
I need some help in understanding the reasoning and analysis in the solution to Exercise 4.5 in Robert C. Wrede and Murray Spiegel's (W&S) book: "Advanced Calculus" (Schaum's Outlines Series).
Exercise 4.5 in W&S reads as follows:
https://www.physicsforums.com/attachments/3918
The...
Now ℝxℝ≅ℂ, seen by the map that sends (a,b) to a + bi. ℂ is a field, so the product of any two non-zero elements is non-zero. However, this doesn't seem to hold in ℝxℝ, since (1,0) * (0,1) = (0,0) even though (1,0) and (0,1) are non-zero. What am I missing?
Also, the zero ideal is maximal in ℂ...
Homework Statement
The vectors that are perpendicular to (1,1,1) and (1,2,3) lie on a ____.
Homework Equations
The Attempt at a Solution
This is really straight forward, but I cannot validate the answer to myself. The textbook says that they should lie on a line, but why is this? Obviously if...
Is it a must to know clifford algebra in order to derive the dirac equation?
I recently watch drphysics video on deriving dirac equation and he use two waves moving in opposite directions to derive it, without touching clifford algebra. If this possible, what is the intuition behind it?
<<Mentor note: Please always use descriptive thread titles.>>
First of all, the title is such that it attracts most views.You see, in class our professor did some goofing around numbers and variables in the relativistic energy momentum relation:
E2=(pc)2+m02c4
Since the energy required to...
1. Are uncountable unions of sigma algebras on a set X still a sigma algebra on X?
2. Are uncountable intersections of sigma algebras on a set X still a sigma algebra on X? (I think this statement is required to show the existence of sigma algebra generated by a set)
3. If 2 is true, can we...
Hello loves, I know there are post very very similar to this, but I just needed some help understanding it. any help is truly appreciated.
1. Homework Statement
A model rocket is launched straight upward with an initial speed of 50 m/s. It accelerates with a constant upward acceleration of 30...
HI all,
I am working my way slowly through some classical mechanics books, and I don't think I could function without my CAS (most often my trusty HP 50g, sometimes Giac/Xcas).
I have never been very good at pages of algebra, often dropping negatives or whatever, bad enough that working...
Entering my third year of my bachelor of science majoring in maths/physics and having some trouble deciding what courses to do this semester. I know for sure I will be taking complex analysis and 3rd year quantum however am having trouble picking between 3 in particular for my final two courses...
Is there anywhere that teaches tensor analysis in both tensor and non tensor notation, because I'm having to pause each time i look at something in tensor notation and phrase it mentally in non tensor notation at which point it becomes staggeringly simpler. Any help apreciated