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'm currently an applied math major. I'm creating a schedule for my next semester and I have the choice to take either complex variables or vector analysis with linear algebra and a college geometry course(elective of choice), but I don't know which pairing will be less stressful. I am currently...
Homework Statement
Express the function Y= (abd + c)' + ((acd)'+(b)')' as the complete disjunctive normal form:
2.1 by applying Boole's theorerm,
Homework EquationsThe Attempt at a Solution
I separated the equations to two terms (T1,T2)
T1= (abd + c)' T2=((acd)'+(b)')'
T1= (abd+c)'...
Bert and Ernie are running around a regular polygon with x sides, all of length 12m. They start from the same point and run in opposite directions. If Bert is twice as fast as Ernie, how far will Ernie have traveled when they meet?
Homework Statement
23. In a ABCD quadrilateral let P,Q,R,S be midpoints of sides AB,BC,CD and DA. Let X be the intersection of BR and DQ, and let Y be the intersection of BS and DP. If ##\vec{BX}=\vec{YD} ## show that ABCD is a parallelogram .
Homework Equations
## (\vec{a}\cdot\vec{b})=0##...
Hello! Is there any rule to do sums and products like the one in the attached picture (Lie.png) without going through all the math theory behind? I understand the first (product) and last (sum) terms, but I am not sure I understand how you go from one to another.
Thank you!
Hello! Is there any rule to do sums and products like the one in the attached picture (Lie.png) without going through all the math theory behind? I understand the first (product) and last (sum) terms, but I am not sure I understand how you go from one to another.
Thank you!
Homework Statement
I am trying to solve for x in this equation:
Homework EquationsThe Attempt at a Solution
I tried solving it like a quadratic equation.
This is the most simplified I could get it.
Can someone please help me find a simpler expression
Homework Statement
I want to show that ## \sum\limits_{n=1}^{\infty} log (1-q^n) = -\sum\limits_{n=1}^{\infty}\sum\limits_{m=1}^{\infty} \frac{q^{n.m}}{m} ##, where ##q^{n}=e^{2\pi i n t} ## , ##t## [1] a complex number in the upper plane.Homework Equations
Only that ## e^{x} =...
Hello PF,
I have just finished my first semester in college and did Calc. 3. Now for the spring semester i have to take differential equations and i have been given the advice that linear algebra comes in handy when dealing with DEs. So can anyone recommend a good introduction for linear algebra...
Homework Statement
$$\lim_{x \to -∞}{\sqrt{x^2 + bx + c} - x}.$$Homework EquationsThe Attempt at a Solution
So in problem 1, once I got to a point where I am to divide by the highest power in the denominator(x) I get something like:
$$\lim_{x \to -∞}\frac{bx+c}{\sqrt{x^2+bx+c}+x}$$
Now what I...
Hi!
First off, I am actually a math / econ major. I hope I'm still welcome here
I am trying to figure out if it's worth it to take both of these courses or just one of them. I have not taken LA before.
Course 1:
Addition, subtraction and scalar multiplication of vectors, length of vector...
Homework Statement
Given an operator ##D(\alpha)=\exp\ (\alpha a^{\dagger}-\alpha^{*}a)## and a function ##g(a,a^{\dagger})##, where ##a## and ##a^{\dagger}## are operators and ##\alpha## and ##\alpha^{*}## are complex numbers, show that
##D^{-1}(\alpha)g(a,a^{\dagger})D(\alpha)=g(a+\alpha...
Homework Statement
Find critical numbers of the function: F(x)=t^3/4 - 2t^1/4
Derivative I got: F'(x)=3/4 t^-1/4 - 1/2 t^-3/4
Homework EquationsThe Attempt at a Solution
I have found the derivative and I understand I must pull out a t in order to find critical numbers, and run across this...
Homework Statement
Let ##g_k = 2cos(k/2)## and ##z=e^{ip(N+1)}## where N is an integer.
There are two simultaneous equations:
##E^2 = (g_k + e^{ip})(g_k + e^{-ip}) = 1 + g_k^2 + 2g_k cos(p) ## [1]
##(1+z^2)E^2 = (g_k + e^{-ip})^2 z^2 + (g_k + e^{ip})^2##[2]...
Homework Statement
X,Y,Z(coordinates)
What function corresponds to the best tunnel shape?
g = 9.8 m/s^2(earth gravity)
Homework Equations
F(x)=Y
G(x)=X^2 in the xy plane
G(z)= sin(X) in the xz plane
H(x)= parabolic sinusoid(X^2 and sin(X) both in the xy plane)
The Attempt at a Solution
I have...
I am following some lecture notes looking at the invariance of Poincare transformation acting on flat space-time with the minkowski metric:
##x'^{u} = \Lambda ^{u}## ##_{a} x^{a} + a^{u} ## [1], where ##a^{u}## is a constant vector and ##\Lambda^{uv}## is such that it leaves the minkowski...
The problem
I am trying to write the equation for the plane on the following form ## ax + by + cz + d = 0 ##
$$
\begin{cases}
x = 1 + s - t \\
y = 2 - s \\
z = -1 + 2s
\end{cases}
$$
The attempt
## s, t ## are the parameters for the two directional vectors which "support" the plane.
$$...
Hi,
I've somehow gone the past year without paying attention to the order of the indicies when one is upper and one is lower i.e. that in general ##g^{\mu}## ##_{\nu}## ##\neq g_{\nu}## ## ^{\mu}##.
A have a couple of questions :
1)
##g^{u}## ##_{v} x^{v}=x^{u}## [1]
##g _{v} ## ##^{u} x^{v}...
Homework Statement
[/B]
example problem: 5 = [(x)(4+x)] / (4-x)
answer: 5
Homework Equations
Unsure what to use.
3. The Attempt at a Solution
Not sure what my professor did, but I thought that if i multiply by the reciprocal of something, I have to balance by multiplying the other side as...
In the representation theory of Lorentz transformations, the words Clifford algebra and Dirac algebra are used interchangeably. However, there is a distinction between the two. Indeed, the Dirac algebra is the particular Clifford algebra ##Cl_{4}({\bf{C}})\equiv Cl_{1,3}({\bf{C}})## with a basis...
Let the generators of the SU(2) algebra be ##\tau_{1}##, ##\tau_{2}## and ##\tau_{3}##.
Consider an ##N## dimensional representation, which means that the ##\tau_{i}## are ##N \times N## matrices which act on some ##N##-dimensional vector space.
Consider the ladder operators...
Problem. Let $K$ be the field obtained from $\mathbf F_p$ by adjoining all primitive $\ell$-th roots of unity for primes $\ell\neq p$. Then $K$ is algebraically closed.
It suffices to show that the polynomial $x^{p^n}-x$ splits in $K$ for all $n$.
In order to show this, it in turn suffices to...
I'm currently reading the paper "Higher Spin extension of cosmological spacetimes in 3d: asymptotically flat behaviour with chemical potentials in thermodynamics"
I'm looking at equation (3) on page 4. I know that symmetrization brackets work like this
A_(a b) = (A_ab + A_ba)/2. However I have...
The question I'm stuck on is:
P = NKBT/(V-Nb) - aN2/(V2) -----> (1)
Re-arrange variables in the Van Der Waals equation of state, Eq. (1), so that V always appears in the equation as V/(3Nb) and P appears as 27b2P/a. Then T should appear in the combination 27b kBT/(8a). Call these...
The problem
I have been trying to solve a long problem but my answer differs from my books answer with just a few peculiar terms.
My answer:
##x_1' \vee x_0'x_1x_2 \vee x_0x_1'x_2' \vee x_2##
Book:
##x_1' \vee x_2 ##
My question is:
Is it possible to simplify ##x_0'x_1x_2 \vee...
Homework Statement
Use the definition of exclusive or (XOR), the facts that XOR commutes and
associates (if you need this) and all the non-XOR axioms and theorems you know
from Boolean algebra to prove this distributive rule:
A*(B (XOR) C) = (A*B) (XOR) (A*C)
Homework Equations
All the...
The problem
This is not a complete homework problem. I am at the last step of the solution to a long problem and only interested to know whether these following expressions are equivalent.
My answer:
## a \oplus ab \oplus ac ##
Answer in my book:
## a \oplus b \oplus c ##
The attempt
I...
Homework Statement
This question has two parts.
There is a quarter or radius R that is charged with a net force of Q. A point like charge of net charge q,is at a distance z from the center of the quarter.
Q1: Under what condition could we use Coulomb's law to find the magnitude the force...
The problem
I am trying to show that ##a'c' \vee c'd \vee ab'd ## is equivalent to ## (a \vee c')(b' \vee c')(a' \vee d) ##
The attempt
## (a \vee c')(b' \vee c')(a' \vee d) \\ (c' \vee (ab'))(a' \vee d)##
The following step is the step I am unsure about. I am distributing the left...
I'm looking at purchasing Algebra (2nd Edition) by Michael Artin, is this a good book to purchase as my first intro to linear algebra book for self learning?
So anyone of you know a book that provides a gentle and quick refresher for linear algera, in the spirit of the book "Quick Calculus" by Kleppner and Ramsey?
Now that I am studying quantum mechanics, I feel I need to review the linear algebra I studied during my engineering degree.
Thanks.
0.0162 * 0.000000000000002 divided by 0.000054 * 0.02
I keep getting 0 but when I try to see if its correct, it tells me its not. What the heck is the answer?
#1. How many classrooms would be necessary to hold 1,000,000 inflated balloons? (Assume one balloon is about 1 ft3 and a typical classroom is about 30 ft × 45 ft × 15 ft. Round your answer to the nearest number of classrooms.)
#2. Approximately how high would a stack of 1 million \$1 bills be...
Homework Statement
Hi,
Just watching Susskind's quantum mechanics lecture notes, I have a couple of questions from his third lecture:
Homework Equations
[/B]
1) At 25:20 he says that
## <A|\hat{H}|A>=<A|\hat{H}|A>^*## [1]
##<=>##
##<B|\hat{H}|A>=<A|\hat{H}|B>^*=## [2]
where ##A## and ##B##...
Homework Statement
Construct a subset of the x-y plane R2 that is
(a) closed under vector addition and subtraction, but not scalar multiplication.
Hint: Starting with u and v, add and subtract for (a). Try cu and cv
Homework Equations
vector addition, subtraction and multiplication
The...
This is a linear algebra question which I am confused.
1. Homework Statement
Prove that "if the union of two subspaces of ##V## is a subspace of ##V##, then one of the subspaces is contained in the other".
The Attempt at a Solution
Suppose ##U##, ##W## are subspaces of ##V##. ##U \cup W##...
Homework Statement
define the time in an analog clock up to the second. The time needs to be such that the minute hand, is at the exact same position as the hour hand. This position of the hands needs to be between 10th and 11th hour.
Homework Equations
3. The Attempt at a Solution
[/B]
I...
the answer key said d is supposed to be 10. but there's a way to evade that row exchange. 1st picture is the question and the 2nd picture is the elimination steps.
Ok so a bit of background about myself
I finished
1. Algebra 1 and 2
2. Trigonometry (Basics)
3. Basics of complex numbers
4. The very basics of matrices (addition, subtraction, multiplication, identites, inverses only)
5. Basic differential Calculus
I want to learn mathematics in this order...
I am asking this question as I have found Boolean Algebra quite intriguing. I have a good understanding of high school level probability and statistics and also Algebra II. Is this enough or do I need more "mathematical maturity"? Anyway, thank you in advance.
Hello! I just started reading about SU(2) (the book is Lie Algebras in Particle Physics by Howard Georgi) and I am confused about something - I attached a screenshot of those parts. So, for what I understood by now, the SU(2) are 2x2 matrices whose generators are Pauli matrices and they act on a...
Hello! I am sorry that this questions is not actually directly related to physics, but, can anyone recommend me a good book about abstract algebra (basically lie algebra, representation theory etc.) used in physics? I have tried for a long time to find something online but I haven't find a...
Homework Statement
Note: I'm saying it's very very hard because I still couldn't solve it and I've posted it in stackexchange and no answer till now.
I'm posting here the problem statement, all variables and known data in addition to my solving attempts. Because I'm posting an image of my...