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.
Most of the results on google happily prove A+(B.C) = (A+B).(A+C), which is that OR is distributive (over AND).
But as part of their proof, they use the law that AND is distributive (over OR), namely that
A.(B+C) = (A.B)+(A.C) which I can't seem to find any algebraic proof for.
So are there...
This is such a basic question that I'm embarrassed to even ask it but I don't really know where else to turn. I was reading through a derivation and one of the steps goes from \frac{\sqrt{a}}{n} = \frac{\sqrt{A}}{n+m}
to
n = \frac{\sqrt{a}*m}{\sqrt{A}-\sqrt{a}}
I'm getting really...
In Bruce Cooperstein's book: Advanced Linear Algebra, he gives the following example on page 12 in his chapter on vector spaces (Chapter 1) ... ...I am finding it difficult to fully understand this example ... ...
Can someone give an example using Cooperstein's construction ... using, for...
I'm grading for a linear algebra class this semester. The class is comprised entirely of engineering majors of various flavors. The homework assigned by the professor is almost entirely "proofs" they are fairly specific proofs. Really the only thing that designates them as proofs is that the...
Homework Statement
The 1.1m bar is supported by a ball and socket support at A and 2 smooth walls. The tension in the vertical cable CD is 1kN.
Draw the Free Body Diagram of the Bar.
Determine the Reactions at A and B. Homework Equations
Equillibrium equations.
Perhaps the vector...
Homework Statement
Let (S,\cdot ) be an algebraic structure where the operation \cdot is associative and commutative and also the following axiom is satisfied:
\forall x,y\in S,\exists z\in S: zx = y\ \ (1)
Prove that if for every a,b,c\in S, ac = bc, then a=b.
Homework EquationsThe Attempt...
Homework Statement
Am I doing this right? The question is...
(ab) + (a' + b') = 1
Homework Equations
(a) Commutative a · b = b · a a + b = b + a
(b) Associative (a · b) · c = a · (b · c)...
Homework Statement
Consider a semi simple lie algebra. Show that if ##T_a## are the generators of a semi simple Lie algebra then ##\text{Tr}T_a=0##.
2. Homework Equations
The commutation relations of the generators and the cyclic properties of the trace.
The Attempt at a Solution
$$[T_a...
Say I have the algebra \mathbb{Z}_5[x]/I where is the I is the principle ideal generated by x^2+4. How do I find the ideals in A? I can't seem to find an explanation that is clear anywhere. Thanks!
Homework Statement
Let ##x,y,z## be arbitrary integers and the GCD: greatest common divider between any two is ##1##. Let ##(G,\cdot )## be abelian. Prove the implication:
a,b\in G\left (a^x = b^y = (ab)^z = e\Longrightarrow a=b=e\right )
where ##e## is the unit element of ##G##.
Homework...
Hi, I am looking for a book that explains tensors and builds a working knowledge of tensors, like the book Linear Algebra by Friedberg Insel and Spence, which I thought explained things very well (if you haven't heard of it, its an intro. book on linear algebra). Thanks!
Homework Statement
Let A be the algebra \mathbb{Z}_5[x]/I where I is the principle ideal generated by x^2+4 and \mathbb{Z}_5[x] is the ring of polynomials modulo 5.
Find all the ideals of A
Let G be the group of invertible elements in A. Find the subgroups of the prime decomposition.Homework...
Homework Statement
It is a potential barrier with E of the incoming matter wave E>U (greater than the height of the barrier). I have already done a LOT of algebra to get to the point where
##\frac{F}{A}=\frac{2kk'e^{-ikL}}{2kk'cos(k'L)-i(k'^{2}+k^{2})sin(k'L)}##
which I won't bother writing...
Hello folks! New to this forum, so hoping I'm not retreading old ground. The Pauli matrices are spin angular momentum operators in quantum mechanics and thus are axial vectors. But in Clifford algebra in three dimensions they are odd basis elements and thus polar vectors. Hestenes, Baylis, other...
Homework Statement
There is an equation in my class notes that I do not know how to solve. It is basically transforming equation A to equation B. It is very important to me that I know how to transform it.
Homework Equations
Attached as a picture.
The Attempt at a Solution
I do not know how...
Hey guys,
So I'm stuck on another question from the previous one that I posted and would absolutely love it if I can get some help regarding how to attempt this. I literally have no clue at how to go by solving it. I have a feeling for question one that the cosine laws might come in handy but...
Hi,
I have been wanting to do this for a while but not too sure how to go about it. I have the following geometric algebra
\lbrace\mathbf{e}_{i}\rbrace_{i=0}^{3} which satisfy the following relations: \mathbf{e}_{i}\mathbf{e}_{j}=-\mathbf{e}_{j}\mathbf{e}_{i} and...
Hi, say we have entities A,B,C in a Relational Database. Is there a sort-of operational algebra that allows us to determine that if, say, A and B are in a many-many relation and B and C are in a 1-many relation, then A,C are in a certain relationship, i.e., a composition algebra between...
Homework Statement
If an investment increased by 40% over 3 years and rose by the same percentage rate each of those years, what was the rate?
Homework Equations
new_value = old_value(1+r)^p
The Attempt at a Solution
Let value of the investment be 1000, then we have 1000(1+0.4) = 1000(1+r)^3...
Homework Statement
Good day,
I need to show that S_n=\mathbb{Z}_2(semi direct product)Alt(n)
Where S_n is the symmetric group and Alt(n) is the alternating group (group of even permutations) note: I do not know the latex code for semi direct product
Homework Equations
none
The Attempt at...
Hi, I'm currently studying to become a chemical engineer.
After learning differential equation and linear algebra, I've realized how useful they are in my engineering courses since they make setting up equations and solving them so much easier. So I was wondering if there are other math that...
Homework Statement
I solved the problem myself but i have a question about the algebra
Homework Equations
n(A ∪ B ∪ C) = n(A) + n(B) + n(C) -n(A ∩ B) -n(A ∩ C) -n(B ∩ C) + A ∩ (B ∩ C)
= n(A ∪ B) -n(A ∩ C) -n(B ∩ C) + A ∩ (B ∩ C)
The Attempt at a Solution
I knew i needed...
Homework Statement
Construct a 3x3 matrix A and vectors b and c in R^3 so that Ax=b has a solution but Ax=c
Homework EquationsThe Attempt at a Solution
So I don't know where to start. I am not sure if the problem is asking me to create a matrix with real numbers or variables. What I do know is...
Homework Statement
a. Lucy's Lunch and Latte has found that customers are put off by the local tourism tax of 9% that is added to their bill. If Lucy decides to cover the tax herself, rather than adding it to the customer's bill, what percent will the customer see in savings?
b. Lucy decides...
So, I'm trying to think of some way to formulate a "Conservation of Symbols" Law of Mathematics. Something like this:
By "symbol" I mean any atomic variable, constant, digit, operator, bracket, etc., that is present in a syntactically correct expression, equation, or inequality. So the...
This is only a minor question.
I watched an on-line video recently on su2 and how it applies to Physics. Now, one of the first things the instructor did was to change the base to sl2. Fine and all, but she called sl2 "su2" for the whole video. Since the two Lie algebras have different...
Homework Statement
Prove that if ({A_1, A_2, ..., A_k}) is a linearly independent subset of M_nxn(F), then (A_1^T,A_2^T,...,A_k^T) is also linearly independent.
Homework EquationsThe Attempt at a Solution
Have: a_1A_1^T+a_2A_2^T+...+a_kA_k^T=0 implies a_1A_1+a_2A_2+...+a_kA_k=0
So...
So basicly our teacher taught us in high school how to find the product of some equations but I do not understand it very well and I need someone to teach me how to solve this basic problem.
The Equation is : (3x^2-4x+1)(4x^2+x-2)
I do not know how to find the product of that problem can...
Homework Statement
Determine if the following sets are bases for P_2(R)
b) (1+2x+x^2, 3+x^2,x+x^2)
d) (-1+2x+4x^2, 3-4x-10x^2,-2-5x-6x^2)
Homework Equations
Bases IF Linear Independence AND span(Set)=P_2(R)
RREF = Reduced Row Echelon Form
The Attempt at a Solution
My first question here...
Dear all, Can someone please recommend a english version of a linear algebra book. "Lineare algebra, Siegfried Bosch" I don't speak German so I cannot understand the chapters. If someone can help to identify a book with similar chapters that will be very helpful.
you can find the full book...
It's rather strange, isn't it? I did just fine in Algebra I, and even better in Algebra II, but now that I'm in Calculus I I'm finding that my algebra skills aren't as sharp as I thought they were. Often times i'll get the calculus portions of problems correct, but when it comes down to...
I have a question about Automorphisms. Please check the following statement for validity...
An automorphism of a group should map generators to generators. Suppose it didn't, well then the group structure wouldn't be preserved and since automorphisms are homomorphisms this would be a...
In quantum mechanics, a physical quantity is expressed as an operator G, then the unitary transformation coresponding to the physical quantity is expressed as exp(-iG/ħt), being also an operator, where t is the tranformation parameter. G is actually the conservative quantity corresponding to the...
Homework Statement
True or false?
If T: ℙ8(ℝ) → ℙ8(ℝ) is defined by T(p) = p', so exists a basis of ℙ8(ℝ) such that the matrix of T in relation to this basis is inversible.
Homework EquationsThe Attempt at a Solution
So i think that my equations is of the form:
A.x = x'
hence A is...
What are some books that I can use master DiffyQ and linear algebra ? I want to improve significantly in E&M and quantum mechanics. I understand that I have to practice, practice, and practice , but can you provided me with with some advice ? Thank you.
Hello there,
I was wondering if someone might be able to help me out with some intermediate steps please. I can't see how
$$f_{tt} =-\int_{r_0}^0 \left(\frac{2Gm_0}{r} - \frac{2Gm_0}{r_0}\right)^{-\frac{1}{2}} dr$$
becomes
$$f_{tt} =-...
I don't even know what is the general name for the thing whose specific name I am trying to find out, but in an algebra question, say 3x = 6y, there is the "assumption" that by dividing both side by 3, we will get x = 2y and this equation is as correct as the original one. What is thing that...
So, I have an equation:
~A * B * C * ~D + ~A * B * C * D + A * ~B * ~C * D + A * ~B * C * D + A * B * ~C * D + A * B * C * ~D
where * represents "AND" and + represents "OR", ~ being NOT.
Part of the reason I'm having trouble is due to the length of the equation.
So far, I've managed to use...
Hello all,
I am about a week into a quantum mechanics course in which the instructor is mostly going to follow along with Sakurai's Modern Quantum Mechanics. However, my linear algebra is pretty rusty and I have never taken an actual course in linear algebra, I have just learned what I have...
Homework Statement
Suppose A is non-singular (nxn) matrix. Given that AB=(zero matrix), show that B=(zero matrix).
Hint: Express A as [A1, A2, ..., An]
and Express AB as [AB1, AB2, ..., ABn]
Homework EquationsThe Attempt at a Solution
[/B]
I argued that because A is non-singular, A=[A1...
Hi all,
Son came home with these questions and is unable to solve them.
I am also struggling with these questions. I need to explain to my son how to get the solutions and not just the answers.
Any help would be greatly appreciated!
1. Two cars are traveling along two straight roads which...
My professor was proving that the Dodecahedron is isomorphic to ##A_5## and in the process utilized the stabilizer (which one can intuit ) of an edge, vertex or face to determine the conjugacy class (which is hard to intuit) of elements of the same order. This seems like a valuable skill but I...
So I stumbled upon this problem:
Solve:
And I have attempted to solve it, however my solution doesn't match that of the book.
The solution should be x+4, but I don't have a clue how to get to that. I'm sorry if this seems pretty rudimentary, but I just want some help because it's the first...
Homework Statement
The Tensions in four cables are equal:
|T1| = |T2| = |T3| = |T4| = T
Determine the value of T so that the four cables exert a totl force of 12,500-lb magnitude on the support.
(Inser a picture of 4 connected cables angled 9, 29, 40, 51 degrees respectively)
My question...
https://www.physicsforums.com/threads/derivative-of-1-x.211489/
In the last post, three last steps. He changes -h into -1, how does this work? I know that if you for example have h/h it's equal to 1 regardless of what value h has, since they're both the same value. I don't see that being...
I am preparing myself for a BSEE starting next year. I have just finished Stewarts Calculus for Calc I,II,III and Diff Eqs up to solving 2nd order non-hom diff eqs by undet. coeffs, variation of parameters and power series.
In high school some 15 years ago I saw some very basic linear algebra...
Homework Statement
Suppose a linear transformation T: [P][/2]→[R][/3] is defined by
T(1+x)= (1,3,1), T(1-x)= (-1,1,1) and T(1-[x][/2])=(-1,2,0)
a) use the given values of T and linearity properties to find T(1), T(x) and T([x][/2])
b) Find the matrix representation of T (relative to standard...
Homework Statement
What is the volume of the parallelepiped with vertices (3, 0, −1), (4, 2, −1), (−1, 1, 0), (3, 1, 5), (0, 3, 0), (4, 3, 5), (−1, 2, 6), and (0, 4, 6)?
Homework Equations
The volume of a parallelepiped is given by the triple scalar product, (a × b) ⋅ c
The Attempt at a...
Right now I'm using "Affine Lie Algebras and Quantum Groups" by Fuchs. I'm getting sick of it. As a Physicist the structure and focus of the text is attractive to me. But, to give one example...
The text constructs two vector spaces: L and Lw. Lw is dual to L. The text goes on to state...