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.
http://arxiv.org/abs/1001.2485
---
The above paper is about a possible two-time formulation of physics. It is by serious people.
To understand it I'm trying to generalized the Dirac eqn. to 3+2 dimensions with signature (++---)
I found the following (now closed post) useful...
I was wondering if there was anyone comfortable enough about basic to advanced linear algebra concepts who would be willing to participate in a chat/whiteboard based platform to discuss or answer questions concerning such concepts. I need clarification on a few things
Homework Statement
If ##a+\frac{1}{a}=\sqrt{3}##, then ##a^{2016} + (\frac{1}{a})^{2016} ## equals
A. 0
B. 1
C. √2
D. √3
E. 2
Homework EquationsThe Attempt at a Solution
[/B]
I find no real solution for a.
What I can do is just find a^2 + (1/a)^2 which equals 1, a^4 + (1/a)^4 = -1, a^8 +...
I've been learning algebra for the past 2 years (in high school), not once have we ever had to rationalize a denominator in a radical expression. I am now relearning Algebra and Trig., what use is there? I mean, all you're doing is switching the numerator (rational) to the Denominator...
Homework Statement
I am watching a course on Relativistic Quantum Mechanics to freshen up, and I have found to have some issues regarding simple operator algebra. This particular issue on the Pauli Equation (generalization of the Schrodinger equation that includes spin corrections) in an...
Homework Statement
Prove the following: Let V be a vector space and assume there is an integer n such that if (v1, . . . , vk) is a linearly independent sequence from V then k ≤ n. Prove is (v1, . . . , vk) is a maximal linearly independent sequence from V then (v1, . . . , vk) spans V and is...
Homework Statement
Prove the following theorem: Let (v1, . . . , vk) be a sequence of vectors from a vector space V . Prove that the sequence if linearly dependent if and only if for some j, 1 ≤ j ≤ k, vj is a linear combination of (v1, . . . , vk) − (vj ).
Homework EquationsThe Attempt at a...
Computer science major here.
I recently completed calculus I, II, III. Now I know that I'll probably be required to take linear algebra.
So far, I expect to do some Gaussian elimination (assuming I know what a row echelon is).
What else should I expect from linear algebra?
Hello. I am a sophomore and I attend an Early College High school. I am currently in Geometry this year and I will be taking Algebra II during the first half of summer. To account for the second half-credit of Algebra II, I will be taking College Algebra in the second half of summer. Throughout...
I'm having trouble getting started on this one and I'd really appreciate some hints. This question comes from Macdonald's Linear and Geometric Algebra book that I'm using for self study, problem 2.2.4.
Homework Statement
Let U1 and U2 be subspaces of a vector space V.
Let U be the set of all...
Homework Statement
Solve the equation for r,r>0,r<R.
\frac{-2\pi R^3}{3}-\frac{8\pi r^2\sqrt{R^2-r^2}}{3}+2r^2R+\frac{2\pi}{3}R^2\sqrt{R^2-r^2}=\frac{2\pi R^3}{3}
2. The attempt at a solution
After factoring,
-2R(2\pi R^2-3r^2)=2\pi\sqrt{R^2-r^2}(4r^2-R^2).
After squaring,
64\pi^2...
In the case of two fields interfering with each other when calculating the total electric field, cos (phi1-phi2 + kx) = cos( kx) where kx is the path difference between the two fields.
How does cos (phi1-phi2 +kx)=cos(kx) Isit just algebra?
Homework Statement
Find the SVD of
Homework EquationsThe Attempt at a Solution
I'm stuck
My question is why in the solution it originally finds u_2=[1/5,-2/5]' but then says u_2=[1/sqrt(5),-2/sqrt(5)]'. I don't see what math was done in the solution to change the denominator from 5...
I'm reading Gelfand's algebra, and I encountered some wonky stuff that I can't figure out on my own (hence my need for the book in the first place) in problem 42 of the book. Here is an explicit statement of the problem. I thought parts a. and b. were easy, but when it came to part c., I just...
I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...
I am focused on Section 10.3 The Tensor Algebra ... ...
I need help in order to get a basic understanding of Example 10.1 in Section 10.3 ...Example 10.1 plus some preliminary definitions reads as...
I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...
I am focused on Section 10.3 The Tensor Algebra ... ...
I need help in order to get a basic understanding of Definition 10.5 in Section 10.3 ...Definition 10.5 plus some preliminary definitions reads as...
Hello,
As you might discern from previous posts, I have been teaching myself:
Calculus on manifolds
Differential forms
Lie Algebra, Group
Push forward, pull back.
I am an engineer approaching this late in life and with a deficient background in math. It is all coming together and I almost...
I know the arrow -> means a map.
For example, defines a linear map. But I cannot figure out what does a X mean?
I know that above denote a inner product map. whatever is on the left of the : is defined by whatever is on the right. but what is the x symbol? What is the correct way to read...
Homework Statement
Let A be an n×n matrix which has the property that A^2 =A.
(i) Write down the most general polynomial in AHomework EquationsThe Attempt at a Solution
My biggest problem is that I don't even understand what the question is asking
Is it just sum (alphaA^n)=0
but A^n=A...
Homework Statement
Let(v1, v2, v3) be three linearly independent vectors in a vector space V. Is the set {v1-v2, v2-v3, v3-v1} linearly dependent or independent?
Homework Equations
Linearly independent is when c1v1+c2v2+...+ckvk=0
and c1=c2=...ck=0
The Attempt at a Solution
c1(v1-v2)+...
Homework Statement
I'm told to find the matrix Q of the matrix A
Homework EquationsThe Attempt at a Solution
So my problem is that in the answer key they have S = (1/3)... and I have no idea where this 1/3 comes from. I get an equivalent answer for X_1, X_2, and X_3
S = [X_1, X_2, X_3] but...
which has higher failure rate: linear algebra or intro to programming. I understand failure rate might be skewed: math majors, who are gifted and interested in math, are required to take linear algebra but not intro to programming, while CS majors are required to take both. Which class do CS...
Homework Statement
[/B]
Th value of 'a' for which the equation x3+ax+1=0 and x4+ax+1=0 have a common root is?Homework EquationsThe Attempt at a Solution
i initially thought of subtracting both the equations and then finding x and substituting back in the equation but it did not work.
It is a bit of a long question about series, But what is important is this...
a+ 2c = 3b
(3b-2)^2 = 2a * (4c-2)
It is asking for the ratio c to a
It have tried a lot of ways. I always end up with this. (Note maybe I not noticing a mistake, Can just anyone confirm that this is solvable? )...
Homework Statement
Let R2 have the Euclidean inner product and use the Gram-Schmidt process to transform the basis {u1,u2} into an orthonormal basis.
u1 = (1,-3)
u2 = (2,2)
Homework Equations
Gram-Schmidt process:
\\v_1 = u_1
\\v_2= u_2 -
\frac{\left ( \left \langle u_2, v_1\right \rangle...
where ##□=\nabla^{\mu}\nabla_{\mu}## is the covariant D'Alembertian.
##□x^{\mu}=0##
##g^{\rho\sigma}\partial_{\rho}\partial_{\sigma}x^{\mu}-g^{\rho\sigma}T^{\lambda}_{\rho\sigma}\partial_{\lambda}x^{\mu}=0##
So this line is fine by subbing in the covariant derivative definition and lowering...
Homework Statement
I'm trying to solve this circuit
when I try to simulate and run a dc sweep i get that it can't be solved. When I try to find the answer using linear algebra
I get this answer after I throw it into MATLAB with the warning
“[w]arning: Matrix is close to singular or...
Homework Statement
I feel like I should know the answer to this, so I believe this to be an easy question. Say have matrix A, and I store the elimination matrices E_1,1 E_2,1 etc. and somewhere in the elimination process I have to use a permutation matrix to swap rows. My question is when I...
I am reading in my group theory book the well known commutation relations of the Lie algebra of SO(3), i.e. [J,J]=i\epsilon J.
What I don't understand is the statement that "from the relations we can infer that the algebra has rank 1".
Any ideas?
Homework Statement
So I think I found an error in the solution were it attempts to find q_2^
I'm asked find the orthornomal basis for the column space of matrix A.
Homework EquationsThe Attempt at a Solution
[/B]
My question is in what it puts for q_2^
A_2 = [4/3 4/3 -2/3]^T
||A_2|| =...
Homework Statement
A line is given by the equation ##x + 2y - 3z = 7##.
Find any vector in the direction parallel to this line in the Cartesian coordinate system.
Homework Equations
I imagine that there are some fundamental relationships I am missing here that would make this more...
Homework Statement
I feel like this is a easy question but it seems the answer key doesn't seem to be right.
So say I have 2 vectors
and I'm trying to find the projection of vector u perpendicular to the vector v
Homework EquationsThe Attempt at a Solution
So I don't remember doing...
Homework Statement
I ask for help in solving the exercises in this project on applied linear algebra. The problem outlined in the project is one in which we are tasked with modeling the demise of a gambler.
I need help solving exercise 1 (in red) on page 6. I have pasted the exercise text...
Homework Statement
So yeah I'm doing a project were I get to create a problem. I would like to learn how to solve a AC RLC circuit using linear algebra. I'm trying to find all of the currents on the edges of the graphs and find all of the voltages at the nodes connecting the edges. I don't...
Homework Statement
I got following Question from a book:
I am doing it using a different method but my answer is wrong. Can somebody please guide me, what is the problem with it??
The question is:In 1999, Diana read 10 English books and 7 French books. In 2000, she read twice as many French...
Homework Statement
I'm trying to show that the output of this XOR circuit is ##F=A'B+AB'##,
Homework Equations
##(A+B)'=A'\cdot B'##
##(A\cdot B)'=A'+B'##
The Attempt at a Solution
From the gates the output is ##[(A\cdot B)+(A+B)']'##, using De Morgan's laws this becomes ##[(A\cdot...
I'll try to be concise. I've been out of math for years and never truly learned to understand it. Until now. I want to put the growth mindset theory to the test and see if I can handle physics (or any STEM field) on a university difficulty. To verify if I'm up to it and even have the slightest...
Homework Statement
Let A be an n x n matrix, and let v, w ∈ ℂn.
Prove that Av ⋅ w = v ⋅ A†w
Homework Equations
† = conjugate transpose
⋅ = dot product
* = conjugate
T = transpose
(AB)-1 = B-1A-1
(AB)-1 = BTAT
(AB)* = A*B*
A† = (AT)*
Definitions of Unitary and Hermitian Matrices
Complex Mod...
Hi
I'm an advanced undergraduate physics student and I'm currently searching books for my career project.
The topic I selected is titled: Algebra and geometry in modern physics.
So I'm currently looking for books that cover modern aspects of physics in a more mathematical approx.
In specific I'm...
This article sounds or reads like the same old problem: http://news.yahoo.com/algebra-unnecessary-stumbling-block-us-schools-170616735.html
Why should that be? Students fail algebra 1 either because of lack of effort, lack of motivation, or, if for any student the course is difficult, the...
Hi everyone, I am new here. I know this has been asked before so forgive me, but ima ask again. :)
So I'm n pre cal right now and it's going fairly well. I think I'll pass with a B. Next I need to take calculus 1..then later calc 2. So I have 2 options. 1. I can take calculus 1 this summer. I...
Hello
Could someone suggest a good book for my son to read to learn algebra?
I look in the bookstores and the books titled "Algebra..." all involve equations and unknowns and arithmetic manipulations and train or work problems (2 farmers plow three fields in one day, etc)
I am interested in...
Does taking calculus based physics make it fun and more understandable, compared to algebra based physics? It seems like the algebra-based physics you don't go in-depth.
Homework Statement
In West Cornwall, I met w women each carrying s sacks, each containing c cats. However, a third of the women then lost half their sacks, and the remaining women lost half their cats from a third of their sacks. How many cats remained, it terms of w, s and c?2. The attempt at...
I'm trying to prove that kinetic energy is not conserved in inelastic collisions using the conservation of momentum. This is the set-up. An object A of momentum ##{m_1}{v_1}## collides inelastically with object B of momentum ##{m_2}{v_2}##
using momentum conservation ##P_i = P_f##
{m_1}{v_1} +...