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.
Homework Statement
Let V be the set of all ordered pairs of real numbers. Suppose we define addition and scalar
multiplication of elements of V in an unusual way so that when
u=(x1, y1), v=(x2, y2) and k∈ℝ
u+v= (x1⋅x2, y1+y2) and
k⋅u=(x1/k, y1/k)
Show detailed calculations of one case...
Homework Statement
From Linear Algebra with applications 7th Edition by Keith Nicholson.
Chapter 9.2 Example 2.
Let T: R3 → R3 be defined by T(a,b,c) = (2a-b,b+c,c-3a).
If B0 denotes the standard basis of R3 and B = {(1,1,0),(1,0,1),(0,1,0)}, find an invertible matrix P such that...
Hey! :o
Could you give me some information about differential algebra? What is it about?
Differential-algebraic equations (DAEs) are polynomials with complex coefficients and the unknown variables are $z, x, x'$.
Is this correct?
What is the difference between them and the ODEs?
Two...
Frustratingly although I can solve the ODE, I am getting a different answer to the book. Now going in circles so would appreciate a fresh pair of eyes.
The ODE (for a boat coasting with resistance proportional to $V^n$) starts as $ m\frac{dV}{dt} =-kV^n $ Find V(t) and x(t), V(0) = $V_0$
I...
Hey all. So I have been reading this article and have a question I would like to ask. I will be referring to this article extensively so it would be kind of you to open it: http://www.ee.ucr.edu/~yhua/MILCOM_2013_Reprint.pdf
I believe reading the article is not required to answer my questions...
Homework Statement
Let T:V→V be a linear operator on a vector space V over C:
(a) Give an example of an operator T:C^2→C^2 such that R(T)∩N(T)={0} but T is not a projection
(b) Find a formula for a linear operator T:C^3→C^3 over C such that T is a projection with R(T)=span{(1,1,1)} and...
My situation is that I'm 17, and graduated high school early due to the CHSPE test. The only math I have any real knowledge of is algebra 1, since I forgot geometry and barely got into algebra 2 before I graduated. No books have worked, Gelfand's Algebra frustrated me exceptionally because it...
Homework Statement
First off i wasn't sure if i should put this in precalc or here so i just tossed a coin[/B]
I must find the roots of the expression z^4 +4=0 (which I've seen repeatedly on the internet)
Use it to factorize z^4 +4 into quadratic factors with real coefficients
The answer is...
I just finished up Stewart's Calculus Textbook, and the last section was on solving 2nd Order Non-Homogeneous Diff Eqs using power series.
I've looked through Paul's Calculus page in the Differential Sections, and can see that there is still a lot more beyond Stewart's that I'd like to study...
The state of arithmetic today is disgusting. The textbooks on it are absolutely repelling, the authors treat it like a subject that will be of concern to only babies. They don't show any love, they treat the subject like a dirty rug. It's been two years since I majored in mathematics, since...
can you recommend a good book on complex analysis? I would like a book that can sharpen my skills in solving complex number problems through graphs and also improve the algebraic part like solving problems related to roots of unity etc.
(I have studied calculus myself. I have done a lot of self...
I need a book that is under 400-500 pages which covers all of basic physics nicely. Every introductory algebra-based physics book I have seen has over 1000 pages and I just need a concise book right now, for some personal reasons.
It should cover the following topics:
1. General physics
1.1...
First it starts as
r= p* (50K^-.5 100^.5)
then K=[(50p100^.5/r]^2
So how does the power of 2 get there in the second part when moving K to the other side?
i find here a representation of the Lorentz algebra.
Starting from the matrix representation (with the ##\lambda## parameter) i see
how one gets the matrix form of ##iJ_z##
I am less comfortable with the ## -i y\partial_x + x \partial_y## notation
Where does it come from? They say that it is a...
What are some rigorous theoretical books on mathematics for each branch of it? I have devised a fantastic list of my own and would like to hear your sentiments too.
Elementary Algebra:
Gelfand's Algebra
Gelfand's Functions & Graphs
Burnside's Theory of Equations
Euler's Analysis of the...
Does anyone know of any gentle, introductory books to LA that assume little prerequisites, even in the way of vectors and matrices? I want something that will give intuition and reasonable proofs, and will provide enough background for something like computational neuroscience. I do not know...
What is a good book on algebra that is highly theoretical and covers functions, the binomial theorem, sequences/series (basically all algebra topics in college) and if possible, elementary number theory?
Most books on high school-college algebra today don't cover theory at all and when they do...
Hello! On this Fall Semester, I will be taking the Linear Algebra with Proofs (Friedberg), Analysis I (Rudin-PMA) or Abstract Algebra I (Dummit/Foote), and Combinatorics (Brualdi). Since I can take only one from Analysis I or AAI on Fall, I am trying to figure out if the introductory...
Dear Physics Forum advisers,
Today, I got two gifts from my research mentor: "Topics in Algebra" by I.N. Herstein and "Abstract Algebra" by Dummit/Foote. I am very happy and grateful for his gifts, but I already have been studying the abstract algebra through Michael Artin and Hoffman/Kunze...
Hi!
I would like to know if my assumptions are right:
Topology is the merging domain of analysis and algebra;
Relational algebra use topological operators;
Relational algebra is a specification of topology
?
Hello folks. I pulled out my algebra and trigonometry book that I kept from college (that I never ended up going). I am brushing up with algebra right now and this is something that stumped me. If you do not mind, I would love to learn how I can think differently in order to complete this...
Is the fundamental theorem of algebra (for polynomials on the complex plane) equivalent to the statement that any polynomial p of degree n>0 can be written
p(z) = c(z - a_1 ) (z- a_2) \cdot \cdot \cdot (z - a_n )
or am I missing some subtle distinction? And if not equivalent, does the theorem...
I haven't gotten any knowledge of physics. I didn't have the opportunity to take it during my high school due to the biology state exam. Will it hurt to take it this semester? Will the calculus based physics be taught the same as general physics (non-calculus based)? If this isn't a good idea...
Homework Statement
Determine whether or not the vector functions are linearly dependent?
u=(2t-1,-t) , v= (-t+1,2t) and they are written as columns matrixes. Homework Equations
Wronskian, but I don't think I should use it because I need to take derivatives so it doesn't seem like it would...
A couple of notes first:
1.
\hom_{A}(-,N) is the left-exact functor I'm referring to; Lang gives an exercise in the section preceeding to show this.
2.
This might be my own idiosyncrasy but I write TFDC to mean 'The following diagram commutes'
3.
Titles are short, so I know that the hom-functor...
Construct a linear system determined by four numbers whose sum is 40, with the first three numbers adding up to 20 and the last three to 30.
a) Explain why this system has infinitely many solutions.
b) Add another condition on the numbers so that a unique solution can be found and then find...
I have a problem with the derivation of the the form of QM starting from the Lie algebra of the Galilei Group like the one given in Ballentine's cap 3.
My Issue is that the procedure is shown almost as unavoidable, And my feeling is that there have to be more postulates that I'm not seeing...
Hi, I'm going to start learning Abstract Algebra, and I was wondering which book, either Lang (his graduate version) or Dummit and Foote, is better. I'm totally okay with terseness to any degree so that isn't an issue for me. Now, I know that Lang is a hardcore graduate book (at least according...
I am having trouble proving that two multivariate formulas are equivalent. I implemented them in MATLAB and numerically they appear to be equivalent.
I would appreciate any help on this.
Prove A = B
A = (Σπ^-1 + Σy^-1)^-1 * (Σπ^-1*π + Σy^-1*y)
y = π+ X*β
Σπ =τ*Σ
Σy = X' * Σβ * X + ΣεB =...
Homework Statement
Q0=Q1+Q2 Q1/C1 =Q2/C2 I substituted it Q1/C1=Q0-Q1/C2 then I got Q1=C1Q0-C1Q1/C2 but the answer has to be Q1=Q0C1/C1+C2 my algebra isn't good enough to solve it what are the steps that I have to take to make it to the answer
Homework Equations
The Attempt...
I am reading Paolo Aluffi's book, Algebra: Chapter 0.
I have a question related to Aluffi's description of monoid rings ... ...
In Chapter III, Section 1.4 on monoid rings, we read the following:
In the above text we find:"Given a monoid (M, \cdot ) and a ring R, we can obtain a new ring...
I can most of the time successfully convert between base 10 and another base or another base and base 10 or between 2 bases where one of them is a power of the other(like base 2 and base 4 or base 3 and base 9).
With negative bases I sometimes don't get what I want in that negative base and...
This isn't homework, I'm just trying to refresh my memory on cyclic groups.
My question is, in this problem solution, how does ##{\sigma_i}^m=1## follow from ##\sigma_i## being disjoint?
Homework Statement
Hello guys
So I have the following problem, given the mapping above I have to check weather it's ring homomorphism, and
maybe monomorphism or epimorphism.
The Attempt at a Solution
So the mapping is obviously well defined, and I have proven it's homomorphism, and it's...
OK, either I'm looking at this the wrong way, or this is way above what a 10 year old should be.
Following are 2 equations containing A, B and C. A solution must be found that solves both equations, i.e. A, B and C are the same in each formula. Also, A, B and C must be whole numbers;
C+2A=3B+2...
Hi please i need help in number 3 of the tutorial questions. It is not an assignment its just a tutorial (read title in the image). I am currently studying for my final and i need help in (3b). the only way I am thinking of solving this questions is to use the equation given in part (d). But...
Dear Physics Forum advisers,
My name is Phoenix, a sophomore with major in mathematics and an aspiring applied mathematician in the theoretical computing. I wrote this email to seek your recommendation on the textbooks for abstract algebra. I want to self-study the abstract algebra during...
I am reading the text Group Theory A Physicist's Survey of Ramond, in particular chapter 7.
He explains classical lie algebra structure using cartan generators and root generators.
He sometimes uses reality condition of structure constant( i think he supposes that all generators are...
Suggest good books to read in an order for one to gain profundity in the particular subject. This may seem to be a rather baffling task, due to the amount of subjects and it's vastness, but if it is done, it could certainly be extremely useful, especially for people like me who are highly...
Hi I'm new here!
I've always loved math and have been really good at figuring out how to solve word problems logically and figuring out math in general. When my teachers explain how to do problems i get confused/can't follow what they're saying but its usually fine because i can just figure it...
Homework Statement
If I have the characteristic equation:
-λ3 + 3λ2 + 9λ + 5
And I'm told that one of its eigenvalues is -1.
How do I find the rest of the eigenvalues?
Homework Equations
-λ3 + 3λ2 + 9λ + 5
The Attempt at a Solution
The furthest I can get is:
-λ3 + 3λ2 + 9λ + 5 = (λ + 1) x...
Hey y'all! I am taking my first linear algebra course next semester and we are using Bretscher's book. I have heard some pretty awful things about the way it is laid out, so I would like to hear your thoughts on the text. For some background information I have completed Calculus 1-3 (using...
I am a Physics major looking into Mathematical Biology (perhaps) in the future, and regardless of where I go, I'm trying to build a solid math background for myself.
I've taken (or plan to take):
Linear Algebra I
Calculus I, II, III, IV
ODE
Non-linear ODE
PDE
Complex Variables (I will...
I was wondering about the following
Λ=I+iT
T are the generators and Λ a continuous LT transformation, thus it is real. Therefore T needs to be imaginary.
And we can find two sets one being the generators for SO(3) J_i and the other for boosts K_i, which are both imaginary.
Now I am wondering...