Algebra Definition and 999 Threads

This isn't for math homework. I'm in self study and came across something in my book that I'm seeking clarification for. The equation: $$0.3\left(50-x\right)=6$$ The solution: $$3\left(50-x\right)=60$$ $$150-3x=60$$ $$-3x=-90$$ $$x=30$$ Simple enough. My question is in regards to this: The...

I heard College Algebra is just a review of Algebra 2 but goes in depth more. I was wondering if it's possible to skip College Algebra if a person has an A in Alg 1 and Alg 2?

Hello! I am reading some representation theory/Lie algebra stuff and at a point the author says "the states of the adjoint representation correspond to generators". I am not sure I understand this. I thought that the states of a representation are the vectors in the vector space on which the...

Once I complete high school geometry, I am planning to take algebra 2 and trigonometry next. I am asking if there are good textbooks out there that provide a full course of algebra 2 and trigonometry. Or should I take these two courses separately? By the way, I am self-studying.

I know both are different courses, but what I mean is, will a proof based Linear Algebra course be similar to an Abstract Algebra course in terms of difficulty and proofs, or are the proofs similar? Someone told me that there isn't that much difference between the proofs in Linear or Abstract...

Homework Statement If A is an n x n matrix over R such that A^3 = A + 1, prove that det(A) > 0 . Homework EquationsThe Attempt at a Solution So, what I've done is factor the expression to get A(A+1)(A-1) = 1, then taking the determinant of both sides, I get det(A)det(A+1)det(A-1) = 1. I...

In my goal to self study up to Calculus, I've utilized the very useful Insights page on what books are best for a real intuitive understanding in basic high school mathematics such as Algebra, Geometry, Trigonometry. I purchased all the recommended books and I was wondering what some of the...

Hello! I am reading so very introductory stuff on geometric algebra and at a point the author says that, as a rule for calculation geometric products, we have that ##e_{12..n}=e_1\wedge e_2 \wedge ...\wedge e_n = e_1e_2...e_n##, with ##e_i## the orthonormal basis of an n-dimensional space, and I...

http://physicstoday.scitation.org/do/10.1063/PT.6.3.20170728a/full/ The chancellor of the California Community Colleges system believes that students who are not majoring in math or science should not have to take intermediate algebra to earn an associate degree. California has the largest...

Homework Statement Consider gamma matrices ##\gamma^0, \gamma^1, \gamma^2, \gamma^3## in 4-dimension. These gamma matrices satisfy the anti-commutation relation $$ \{ \gamma^\mu , \gamma^\nu \}=2\eta^{\mu \nu}.~~~(\eta^{\mu\nu}=diag(+1,-1,-1,-1)) $$ Define ##\Gamma^{0\pm}, \Gamma^{1\pm}## as...

Connections between algebraic structures and geometries are mentioned in almost any course of modern geometry or algebra. There are monographs dedicated to the subject. Unfortunately, the books, I managed to find, are written for professional mathematicians. I am looking for a book that focuses...

IN Srednicki's QFT he seems to make two different choices for normalizing the generators of lie algebras. In chapter 24 (eqn 24.5) he chooses Tr (TaTb) = 2 δab and in chapter 69 (eqn 69.8) he chooses Tr (TaTb) = (1/2) δab Is there a reason for this? Is there any particular reason to make one...

I understand that the standard proof is a bit different from my own, but I want to know if my reasoning is valid. PROOF: Firstly, I assume that x is positive. I then consider p = inf{n∈ℕ : n>x} . In other words, I choose "p" to be the smallest natural number greater than x. If we choose n>p...

Homework Statement Homework EquationsThe Attempt at a Solution the answer for no adjoining _W_W_W_W_W_ for 3 red balls, there are 6 positions so ## 6C_3 = 20## i'm curious, on other way to find arrangement? for adjoining = all arrangement - adjoining all arrangement = 3 red can get to any...

Can someone recommend me some books that have practice problems for algebra with answers ?

Homework Statement Homework Equations f ' (x) = 0 --> to find extreme points The Attempt at a Solution ## 2x^2 + y^2 = 4## ## y = \sqrt{4 - 2x^2}## ## y ' = \frac {-2x} {\sqrt{4-2x^2}} ## ## 0 = \frac {-2x} {\sqrt{4-2x^2}} ## x = 0 also ## 4 - 2x^2 >= 0 ## ## 4 >= 2x^2 ## ## \sqrt{2} >= x...

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 Equations $$ \log_{x} a \ means \ 10^x = a $$ $$ \log_{x} {a^n} = n \log_{x}{a} $$ The Attempt at a Solution $$ \log_{10} {ax} \log_{10}{bx} + 1 = 0 $$ $$ \ means \log_{10} {ax} = 1 \log_{10}{bx} = -1 \ or \ vice \ versa $$ $$ \ for \log_{10} {ax} = 1 \ so \ ax...

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 Homework Equations polynomial equation $$(a+b)^2 = \sum_{k=0}^{n} \binom nk a^{n-k} b^k $$ The Attempt at a Solution i get $$ \frac {1} {32} + \frac {5} {16}\frac {3^{0.5}i} {2} +\frac {10} {8}\frac {3i^2} {4} + \frac {10} {4}\frac {3^{1.5}i^3} {8} + \frac {5} {2} \frac...

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...