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
Homework Equations
f ' (x) = 0 -> to find extreme pointThe Attempt at a Solution
can I differentiate the function directly?
f ' (x) = |1| + | 2x - 2|
0 = 2x - 1
x = 1/2
and x from domain = { 0, 2 }
the range we get :
f(0) = 1
f(2) = 1
f(1/2) = 1/2 + 3/4 = 5/4
min = 1
max...
Homework Statement
Homework EquationsThe Attempt at a Solution
this is the answer
https://www.algebra.com/algebra/homework/Circles/Circles.faq.question.1038060.html
but why the c1 = 0, c2 = 1, c3 = 3, c4 = 6 etc
why not c2 = 2? c4 = 4?
Hey all,
I'm currently working on my CS degree with a mathematics minor. After this Fall, I will only have one more course to take to finish my minor.
I'm debating between Linear Algebra and Deterministic Operations Research. I do have other options, but these seem to be most applicable to CS...
Homework Statement
Homework EquationsThe Attempt at a Solution
5 digits minus 4 digit remaining 11111?
if the 5 digit = 20000
and the 4 digit = 8889
so remaining 11111
digits 1 to 9 have been used?
what am I supposed to do?
Homework Statement
Homework Equations
log_2 x = y
2^y = x
3^2^y
The Attempt at a Solution
log_2 x = y
2^y = x
log_2 {log _3 {log _2 { log_3 {2^y} } } }
what am I suppose to do?
Homework Statement
Linear Algebra Problem: Solving for Euler between two ordered bases
I've got a problem I need to solve, but I can't find a clean solution.
Let me see if I can outline the problem somewhat clearly. Okay, all of this will be in 3D space. In this space, we can define some...
Homework Statement
Homework Equations
cos 2theta = costheta^2 - sintheta^2
The Attempt at a Solution
cos2theta = 1
2theta = 0, 2phi
but i get wrong answer.. how is it?
Homework Statement
Homework EquationsThe Attempt at a Solution
2x - 1 >= 0
x >= 1/2
x - 2 >= 0
x >= 2
for x <1/2
then 2x - 1 and x - 2 are negative
solve : -(2x-1) - (x-2) = 2
-3x + 3 = 2
x = 1/3
for 1/2 <= x < 2
the (2x - 1) positive, (x-2) negative
solve : 2x -1 -x + 2 = 2
x = 1
for...
Homework Statement
Let ##T:M_2 \to M_2## a linear transformation defined by
##T \begin{bmatrix}
a&b\\
c&d
\end{bmatrix} =
\begin{bmatrix}
a&0\\
0&d
\end{bmatrix}##
Describe ##ker(T)## and ##range(T)##, and find their basis.
Homework Equations
For a linear transformation ##T:V\to W##...
Hey everyone! (new to the forum)
I am currently trying to self study more advanced mathematics. I have taken up to multivariable calculus and have taken a class for an introduction to mathematical proofs/logic (sets, relations, functions, cardinality). I want to get a head start on the...
Hi! I recently came across this pic:
Problem is, I have no clue how to tackle this. I didn't find it in any textbooks or anything, so I don't know where to look for it. Sorry if it's a "weird" question, but it's been eating at me for a couple of hours now. I saw it categorized as "high...
Homework Statement
Let T: ℝ² → P² a linear transformation with usual operations such as
T [1 1] = 1 - 2x and
T [3 -1]= x+2x²
Find T [-7 9] and T [a b]
**Though I'm writing them here as 1x 2 row vectors , all T's are actually 2x1 column vectors (I didn't see a way to give them proper...
Homework Statement
Let be T : ℙ2 → ℙ2 a polynomial transformation (degree 2)
Defined as
T(a+bx+cx²) = (a+1) + (b+1)x + (b+1)x²
It is a linear transformation?
Homework Equations
A transformation is linear if
T(p1 + p2) = T(p1) + T(p2)
And
T(cp1)= cT(p1) for any scalar c
The Attempt at...
Homework Statement
Let ##({a, b, c}, *,+)## be a finite field. Complete the field table for the operations ##*## and ##+##
##\begin{array}{|c|c|c|c|}
\hline * & a & b & c \\
\hline a & ? & ? & ? \\
\hline b & ? & ? & ? \\
\hline c & ? & ? & b \\
\hline
\end{array}##
##\begin{array}{|c|c|c|c|}...
About to start reviewing for a math test but it has been ages since I was in school. Hoping some of you math savants could help me out.
Will hope to understand and be able to complete these two exams. Starting with the first one below. (Do you guys know of a better exam to test and know...
Homework Statement
Let { u, v, w} be a set of vectors linearly independent on a vector space V
- Is { u-v, v-w, u-w} linearly dependent or independent?
Homework Equations
[/B]
A set of vectors u, v, w are linearly independent if for the equation
au + bv + cw= 0 (where a, b, c are real...
Hi,
I am a math undergraduate major and just finished my first abstract algebra course. Unfortunately, we used the lecturer's notes which are quite dry, without motivation, and it really felt bad. I am really interested in abstract algebra, and thus has decided to re-learn it over the summer...
I'm reading a textbook that does a couple of algebraic manipulations in one step (in order to get the expressions ready for the binomial theorem). I'm unfamiliar with this step. Could anyone expand it into several steps? It's using some kind of algebraic standard technique or result that I don't...
Hi All,
According to the fundamental theorem of algebra: "every non-zero, single-variable, degree n polynomial with complex coefficients has, counted with multiplicity, exactly n complex roots".
My question is: what about polynomials with degree say 2.3 or 3.02, as in the polynomial:
## p(x) =...
Hello! I read that the for the lie algebra of the Lorentz group we can parametrize the generators as an antisymmetric tensor ##J^{\mu \nu}## and the parameters as an another antisymmetric tensor ##\omega_{\mu \nu}## and a general transformation would be ##\Lambda = exp(-\frac{i}{2} \omega_{\mu...
i have some confusions with some terms ,especially terms like simplify , solve .
Factors ,GCF ,LCM
Should i start learning it like this ?
Simplify fraction
Simplify Algebraic expression
Simplify Algebraic fractions
Solve Algebraic equations
?
I had a pretty tough schedule this semester, so I'm getting my first B. I otherwise wouldn't be too sad, but I hear Linear is pretty important in upper-level physics and astronomy. So, will this hurt my chances of getting into grad school? I am (was? rising sophomore) only a first-year, and I do...
Hi everyone!
I'm having trouble with the following exercise:
Let ##\mathrm {Aff}(ℝ)## be the vector space of the affine maps from ##ℝ## to ##ℝ##:
$$φ_{a,b}:ℝ→ℝ$$ $$x→a x + b$$
Find the contravariant and and covariant coordinate of the map:
$$φ_{1,1}:ℝ→ℝ$$ $$x→x + 1$$ with respect to the...
Homework Statement
Question:
(With the following definitions here: )
- Consider ##L_0|x>=0## to show that ##m^2=\frac{1}{\alpha'}##
- Consider ##L_1|x>=0 ## to conclude that ## 1+A-2B=0##
- Consider ##L_2|x>=0 ## to conclude that ##d-4A-2B=0##
- where ##d## is the dimension of the space...
Homework Statement
Suppose that b is 20% more than a, c is 25% more than b, and d is k% less than c. Find k such that a=d
Homework Equations
a=d; b=1.2a; c=1.25b=1.5aThe Attempt at a Solution
1. a=##\frac{c}{15 }##=d=##\frac{100c-k}{100}##
2. By multiplication of extremes and means we get...
I've always been curious about why we define polynomials the way we do. On the surface, it seems that they are expressions that naturally arise from combining the standard arithmetic operations on indeterminates. However, there are some points that I am generally confused about. Why are...
Homework Statement
Using the power series for ln(x + 1) and the Estimation Theorem for the Alternating Series, we conclude that the least number of terms in the series needed to approximate ln 2 with error < 3/1000 is: (i) 333 (ii) 534 (iii) 100 (iv) 9 (v) 201
Homework Equations
ln(x+1) =...
Homework Statement
In a two-dimensional tug-of-war, Alex, Betty, and Charles pull horizontally on an automobile tire at the angles shown in the picture. The tire remains stationary in spite of the three pulls. Alex pulls with force \vec{F_A} of magnitude 220~N, and Charles pulls with force...
Hello,
I see a lot of people enthusiastic about Clifford algebra and its future role for physics, yet I also see a lot of people frustrated opinions about it. Contents in the internet seems really really small compared with other mathematical topics, not to mention less books about it.
All...
Hi guys,
I am a little confused on the difference between an Algebra and a vector space. I´m guessing there´s a fairly simple distinction. Any guidance would be much appreciated, I accept both hand wavy and hardcore axiom based responses :)
Cheers
How do I go about solving the following equation for n? From inspection, it seems that this equation is not possible since 2n is always positive so positive + 2= 0 is impossible.
2n+ 2 = 0
Homework Statement
Does ##x_p\partial_v\partial_u-x_v\partial_p\partial_u=0##
Homework Equations
I need this to be true to show a poincare algebra commutator.
We have just shown that ##[P_u, P_v] =0 ##, i.e. simply because partial derivatives commute.
Where ##P_u=\partial_u##
The Attempt...
Dear Physics Forum friends,
I am currently stuck with the following question about the integer optimization:
"Produce a family of 0,1 knapsack sets (having an increasing number n of variables) whose associated family of minimal covers grows exponentially with n."
My thought is that I need to...
So I am working out a course schedule for my last two years of undergrad and have room for only one more math class but do not know which would be more beneficial. The two courses are Intro to Modern Algebra or Numerical Linear Algebra. I am working towards a bachelors degree in physics and plan...
I have this integral that when solved, involves squares and natural logs, where ##A\,##,##\,b\,##, and ##\,x_e\,## are constants while ##x## is a variable.
##A = \int_{x_e}^{x} \frac{x^2 - b^2}{x} dx = \int_{x_e}^{x} x \, dx -b^2 \int_{x_e}^{x} \frac{dx}{x} = \frac{x^2}{2} - \frac{x_e^2}{2} -...
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[/B]
How to simplife it? I would like to see a process of simplification.
Y = A'B'C'D + A'B'CD + A'BC'D + A'BCD' + A'BCD + AB'C'D + AB'CD + ABC'D'
The ' denotes a bar over the previous letter.My second question is:
Y = BC + AB'D = (uploaded picture)
Y = A'D + B'D + A'BC +...
Consider the expression
$$\left(T^{a}\partial_{\mu}\varphi^{a} + A_{\mu}^{a}\varphi^{b}[T^{a},T^{b}] + A_{\mu}^{a}\phi^{b}[T^{a},T^{b}]\right)^{2},$$
where ##T^{a}## are generators of the ##\textbf{su}(N)## Lie algebra, and ##\varphi^{a}##, ##\phi^{a}## and ##A_{\mu}^{a}## are numbers.
How...
There is a movie out--"Hidden Figures"--about how three women launched the space program; or something like that.
My daughter wants to write about them for a school project.
I keep suggesting my daughter write about Sophie Germain, or Emmy Noether, Daubechies or Hypatia.
But my daughter...
Homework Statement
http://prntscr.com/eqhh2p
http://prntscr.com/eqhhcg
Homework EquationsThe Attempt at a Solution
I don't even know what these are, it is not outlined in my textbook. I'm assuming I am is image? But how do you calculate image even?
As far as I'm concerned I am has to do wtih...
Homework Statement
I can't understand how abstract aljebra helps in creating graphical patterns. I don't find eq related to Groups. Do we consider predefined structures [/B]
Homework Equations
No equation only patterns. one pattern is attached
The Attempt at a Solution
I don't know how it...
A little background first: As my https://www.physicsforums.com/members/usablethought.611113/ explains, I'm a retired writer/editor, never good at math as a kid but always fascinated by physics, who has decided to take a long, slow, pleasurable road toward learning the necessary math to get to at...
Hello everyone,
After about 6 years I'm finally getting out of the military in May and start going to school full time starting in the fall. I want to do engineering, but I know I'm behind in math and am worried about starting calculus. I want to improve my math by taking both College Algebra...
Homework Statement
Hello there physicsforums,
I have a problem that I'd like feedback on as I'm not sure if I had solved it or approached it correctly.
A ball that is traveling 5m/s is 30 meters above a person who is standing still. How fast must the person travel in order to catch the ball...
https://www.physicsforums.com/threads/discriminant-of-cubic-equation-in-terms-of-coefficients.715480
The thread is really old and didn't want to post this trivial question in there.
The problem is: A cubic equation x³ + px² + qx + r = 0 has three different roots x1, x2, x3.
Find...
Homework Statement
Let ##S = \{\frac{1}{n} + \mathbb{Z} ~|~ n \in \mathbb{N} \}##. I am trying to show that ##f : \mathbb{N} \rightarrow S## defined by ##f(n) = \frac{1}{n} + \mathbb{Z}## is a bijection. Surjectivity is trivial, but injectivity is a little more involved.
Homework EquationsThe...