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.
I want a good linear algebra textbook in order to learn to use linear algebra in physics but also to use it in more theoretical mathematics courses.
I hope that with this poll i will also help others that want to study from a proper Linear Algebra textbook.
I am currently doing aops introduction to algebra book and I can't figure out how to solve joint proportion or some ppl call it joint variation. I don't which values should i substitute for the formula, z=kxy. I don't which value should i substitute into z to the word problems. I can get through...
So last semester, I withdrew from Physics with algebra and trigonometry. I am currently enrolled with another professor and I hope I excel this time in the course. He seems more attentive and friendly, along with having a noticeable passion for Physics. Today was the first day of class and...
Homework Statement
In R^3 with inner product calculate all the least square solutions, and choose the one with shorter length, of the system:
x + y + z = 1
x + z = 0
y = 0
2. The attempt at a solution
So I applied the formula A^T A x = A^T b with A as being the matrix with row 1 (1,1,1)...
A textbook or even better some openly available pdf would be preferred.
I have no idea where to start a search and I normally prefer recommendations over just picking any book from some publisher.
Thank you.
Homework Statement
An electric circuit consists of 3 identical resistors of resistance R connected to a cell of emf E and negligible internal resistance. What is the magnitude of the current in the cell? (in the diagram two of the resistors are in parallel with each other then the other in...
Elements of Lie algebra are generators. So for example Pauli matrices are generators of rotation and the elements of Lie algebra. And multiplication in Lie algebra is commutator. Right?
What about if there is only one generator. As in case in rotation in plane. What is Lie algebra product in...
I am currently in my last year of community college before I transfer to University. Physics held me back! For my major, I am required to enroll in Physics with College Algebra and Trigonometry I & II! I took the initiative to withdraw after completing the second exam because it was extremely...
Homework Statement
Find the peak value of current through the AC source of the following L-C-R circuit, if peak voltage is ##V_0## and angular frequency is ##\omega_0##.
Homework Equations
I have learned Vector algebra and calculus (single variable). I was taught how to use phasor diagrams...
I'm trying to understand why the intersection multiplicity of two singular subvarieties is not equal to the complex dimension of the local ring but it is instead the Euler characteristic.
Is it possible to find an intuitive explanation?
I think that the following concepts need some...
Can you suggest any book for learning multinomial theorem and its application in permutation and combinations problems? I am also looking for a book for learning Permutations and Combinations. (Right now, I am using a problem oriented book by Marcus. But I want a book for learning the basics as...
Dear Physics Forum personnel,
I am currently reading the books called "Linear Algebra Done Right" by S. Axler and "Linear Algebra Done Wrong" by S. Treil. On the next semester, I will be taking the "Second Course in Linear Algebra" which will treat the following topics: determinants...
Homework Statement
Find the span of U=\{2,\cos x,\sin x:x\in\mathbb{R}\} (U is the subset of a space of real functions) and V=\{(a,b,b,...,b),(b,a,b,...,b),...,(b,b,b,...,a): a,b\in \mathbb{R},V\subset \mathbb{R^n},n\in\mathbb{N}\}
Homework Equations
-Vector space span
-Linear independence...
I am reading Steven Roman's book, Advanced Linear Algebra and am currently focussed on Chapter 1: Vector Spaces ... ...
I need help/clarification with respect to a notational issue regarding Roman's definition of the direct product and external direct sum of a family of vector spaces ... ...
This truth table that represents statement p v (q ^ r) is equivalent to (p v q) ^ (p v r)Showing that this statement is not equivalent to (p v q) ^ r.. Now I need to what property of Boolean Algebra is being demonstrated by the fact that the first two statements were equivalentp q r q ^ r...
Intro to Lin. Alg via MIT OCW 18.06 (part of Electrical Engineering), by Gilbert Strang.
I've been doing it as independent study before starting my BSEE next year. I'm getting to the end. Chapter 9 is titled "Numerical Linear Algebra", and is concerned with the heavy, intricate computational...
Hello I have the solution of a problem and I don't understand it
1. Homework Statement
We know that every subgroup L<S10 acts on [1, 10] := {1, 2,..., 10} by the formula π • i = π(i). Consider L the subgroup of S10 generated by the permutation p = (1, 2, 3, 4)(4, 5)(8, 9, 10).
Find the orbit...
Hi y'all,
This is more of a maths question, however I'm confident there are some hardcore mathematical physicists out there amongst you. It's more of a curiosity, and I'm not sure how to address it to convince myself of an answer.
I have a Lie group homomorphism \rho : G \rightarrow GL(n...
Homework Statement
I'm trying to figure out this question:
"Show that the 10-dimensional representation R3,0 of A2 corresponds to a reducible representation of the LC[SU(2)] subalgebra corresponding to any root. Find the irreducible components of this representation. Does the answer depend on...
Homework Statement
Consider the following head-on elastic collision. Particle 1 has rest mass 2mo, and particle 2 has rest mass mo. Before the collision, particle 1 movies toward particle 2, which is initially at rest, with speed u (= 0.600c ). After the collision each particle moves in the...
My Intro to LA course has visited the ideas of polar decomposition and Jordan forms, but not gone into them in depth. I wouldn't say I understood them, but I'm aware of them, and could possibly solve some basic exercises involving them if all I had to do was apply formulas.
My question is...
Homework Statement
1) In a vector space V of all real polynomials of third degree or less find basis B such that for arbitrary polynomial p \in V the following applies:
[p]_B = \begin{pmatrix} p'(0)\\p'(1)\\p(0)\\p(1)\end{pmatrix} where p' is the derivative of the polynomial p.
Homework...
Homework Statement
Check the statement is true or false:
Let \mathcal{A} : \mathbb{R^3}\rightarrow \mathbb{R^4} be a linear operator.
If the minimum rank of \mathcal{A} is 2, than the maximum defect is 1.
Homework Equations
-Linear transformations
The Attempt at a Solution
Assume that...
Hi
Im looking for discrete mathematics and algebra books to study. I want to enhance my understanding of algorithms and programming and i like discrete mathematics
I'm reading Apostol for this semester's calculus (took a look at spivak too).
Any book as good as them that will be useful for me ?
On pages 67 & 68 of Hassani's mathematical physics book, he gives the following definition:
"Let ## \mathcal{A} ## and ## \mathcal{B} ## be algebras. The the vector space tensor product ## \mathcal{A} \otimes \mathcal{B} ## becomes an algebra tensor product if we define the product
##...
Hi All, wondering if this below has been developed already:
There is a context in which adding fractions by the rule : a/b + c/d = (a+c)/(b+d) : say we are considering
the grade in a course where a lot of exams are administered, and the total is considered over, say 1000. Then, if we get 90/100...
Hey guys,
I'm talking to my advisor soon and I was wondering if it is typical to, after taking multi variable calc, to take differential equations and linear algebra simultaneously? I'm going to have to be taking both modern physics and organic chemistry II as well, for context.
Thanks everyone
Hello good people,
please refer to this:
(notice the mistake in 9.31: cos(psi) switches places with cos(phi)sin(psi) to the best of my understanding)
Now, I am trying to derive 9.30 and for this, according to the book, we solve 9.32. The problem is I can not understand 9.32, the meaning of...
I've heard it said that the commutation relations of the generators of a Lie algebra determine the multiplication laws of the Lie group elements.
I would like to prove this statement for ##SO(3)##.
I know that the commutation relations are ##[J_{i},J_{j}]=i\epsilon_{ijk}J_{k}##.
Can you...
Homework Statement
The state of Oregon wishes to design a new lottery game with the following rules: 1.Each ticket costs $5 2.There will be three prizes: $10, $100 and $1000 3.The probability of the $10 prize will be 20%. 4.The probability of the $100 prize will be 1% 5.Ten thousand...
Hi,
I saw a derivation in a book and I don't see the logical connection. Suppose 1.\ \ a=b \text{ and } 2.\ \ x=y
Then 3.\ \ a-b= \lambda (x-y) makes sense to me since 0=\text{Anything}⋅0
However they said "similarly" 4.\ \ a+b= \mu (x+y) , and this I don't understand. To me a+b= 2a...
Homework Statement
How many non-isomorphic groups of two elements are there?
Homework EquationsThe Attempt at a Solution
I don't understand exactly what we are being asked.
If we have a group of two elements under, say, addition, then G =\{0, g\}.
Then also g+g = 0 must be true, means g is its...
Homework Statement
Let T: P2 --> P3 be the transformation that maps a polynomial p(t) into the polynomial (t+5)p(t).
a) find the image of p(t)= 2-t+(t^2)
b) Find the matrix for T relative to bases {1,t,t^2} and {1,t,t^2,t^3}.
Homework Equations
Given
The Attempt at a Solution
a) I know...
Homework Statement
What books of completely solved problems (free in pdf) in linear algebra would you suggest?
Please suggest books that have solved problems, and not theory.
Homework EquationsThe Attempt at a Solution
Homework Statement
[/B]
I am trying to solve the simultaneous inequalities (1) and (2) shown in the following image. The solution is provided, but I'm not sure how they solved for it.
[PLAIN]http://
Homework Equations
N/A
The Attempt at a Solution
[/B]
I tried to solve this set of...
The generators ##(A_{ab})_{st}## of the ##so(n)## Lie algebra are given by:
##(A_{ab})_{st} = -i(\delta_{as}\delta_{bt}-\delta_{at}\delta_{bs}) = -i\delta_{s[a}\delta_{b]t}##,
where ##a,b## label the number of the generator, and ##s,t## label the matrix element.
Now, I need to prove the...
In the figure, ABCD is a square of side 1 cm. ABFE and CDFG are trapeziums(/trapeziods?). The Area of CDFG is twice the Area of ABFE. Let x cm be the length of AE.
(a) Express the length of FG in terms of x.
(b) Find the value of x, correct to 2 decimal places.
Thanks! :D
It is from some famous publications. But I seem can get it from rigorous proof after many hours and different methods of trying and Googling.
If we have g as the eigenvector of a symmetric matrix and G is the eigenvalue of the symmetric matrix.
\left[ {\begin{array}{*{20}{c}} {X11 -...
Homework Statement
Fractions ##\frac{a}{b}## and ##\frac{c}{d}## are called neighbor fractions if their difference ##\frac{ad - bc}{bd}## has numerator ##\pm 1##, that is, ##ad - bc = \pm 1##.
Prove that
(a) in this case neither fraction can be simplified (that is, neither has any common...
I feel as if my understanding of physics require increasingly rigorous mathematical derivations the more advanced the notions get... since I started even high school physics, I lost track of the number of instances where the bulb lighted up in my understanding of physics due to a rigorous...
Homework Statement given two unit vectors a= cosθi + sinθi b=cosΦi+sinΦj prove that sin(θ-Φ)=sinθcosΦ-cosΦsinθ using vector algebra[/B]Homework Equations sin(θ-Φ)=sinθcosΦ-cosΦsinθ[/B]The Attempt at a Solution axb= (cosθsinΦ-cosΦsinθ)k and I'm guessing that the change in sign has...
Homework Statement
The generators of the ##SO(n)## group are pure imaginary antisymmetric ##n \times n## matrices. Therefore, the dimension of the ##SO(n)## group is ##\frac{n(n-1)}{2}##. Therefore, the basis for the so(n) Lie algebra is given by the ##\frac{n(n-1)}{2}## basis vectors as...
I am currently in year 9 (9 grade for those in US) and I have a really rusty and a weak math background. I have 2 months of summer holidays coming up. I should be done with pre algebra in mid December. During my summer holidays I have more than 50 hours a week avalible for study and I was just...
Homework Statement
Let ε = { (-∞,a] : a∈ℝ } be the collection of all intervals of the form (-∞,a] = {x∈ℝ : x≤a} for some a∈ℝ.
Is ε closed under countable unions?
Homework Equations
Potentially De Morgan's laws?
The Attempt at a Solution
Hi everyone,
Thanks in advance for looking at my...
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...