Linear Algebra and Group Theory

Prove that P = a ± b? ABSTRACT. The aim of this proof is to create a system that can solve itself without human intervention which encompasses an AI choice mechanism, a self awareness system and a model for an expanding universe. The AI choice mechanism and the self awareness system is after the model must have been created. Basic number theory shows that it is possible to combine any number by the addition of two other suitable numbers.Still in general, it is not impossible to prove that P = a ± b because the said "Proof P = a ± b" support the infinite list of possible numbers which when added together sum to any given number. The proof: Prove …

It seems like there is some hidden fundamentals to linear algebra that supersedes the systems of linear equations. For example, in DSP, complex exponentials are considered eigenvectors to LTI systems. This is very fascinating, and I would like to know if this is based on systems of linear equations, or there is some way of transferring stuff into linear algebra.

Dimensions. I was browsing Wikipedia the other day and had a look at Dimensions. I now have questions about dimensions that Wikipedia didn't cover. I just want to go through step by step in a slow tedious logical manner and obtain a reasonable answer to my questions. Hope someone can help. Q1. Is the 1st dimension just a perfectly straight line on a plane of infinite length that never bends. Or can it also be a bendy line that curves all over the place in 3 dimensions? Personally I think it can only be a perfectly straight line. But need conformation. Q2. This line, could also never have any width or height it could only have length. Is this true? If it …

Recently, I have come up with an algebraic structure that I would like to call "Partial rings" with some interesting properties. They are similar to rings, as will be shown later. The rules for this structure are: 1) Group under addition 2) Another function, multiplication 3) mutiplication is distributive 4) multiplication is associative 5 (optional)) Noncommutative group under addition. Some properties already found: 1) The identity under addition annihilates under multilplication (clearly) 2) The additive order of the product a*b divides the additive order of each a and b 3) The set of products is commutative 4) The set of products a*r, r is in R is …

Let's think about this problem. This operation @ is consisted of two of multiplying or dividing plus one constant a(a=/ 0). a @ a= a 2 @ 3a = 2/3 2a@ 2 = a^2 1 @ 1 = ? hint @ : ( ) # ( ) # ( ) where ( ) means constant or variables where # means multiplication or division. The first operator searching problem.

Hi everyone, I have to solve the following problem, but I can't find the solution: A is an 12x12 symmetric matrix. Let v be an eigenvector for lambda1. Explain why the power method (in almost all cases) converges to the eigenvalue lambda2, (the eigenvalue of A with the seconde largest absolute eigenvalue), if one starts with a vector x0, which is orthogonal to v. They give the following hint: note that A is symmetric, and write x0 as a linear combination of eigenvectors Thanks a lot for your replies. It would really help me if someone knows how to solve this question. Kind regards, Bob

I am not able to understand clearly the concept of equivalence classes. Is it mandatory that for a particular equivalence relation in Set A, all the equivalence classes are either mutually disjoint or equal? Or are they mutually disjoint for all possible E-relations for a set A(or only on specific E-relation?)? Secondly, In the equivalence relation: Let S be the set of all integers and let n > 1 be a fixed integer. Define for a,b belonging to S, a~b, if a-b is a multiple of n. How is it, in the case above, the equivalence class of a consists of all integers of the form a + kn, where k=0,1,-1,2,-2,3,-3,...; there are n distinct equivalence classes, na…

After several attempts with geometry and gymnastics with number theory, what I believe I found a general proof and simple one as Fermat would wish. But anyway, I am not skilled in Matematics, nor in any subject except chemistry, I just get sometimes the creative ideas.... So I got one last night...I would really like to ask you guyz, where is the mistake?As probably there is one. Proof: Let us suppose that a,b,c are coprimes, so if we construct the from a,b,c the smallest triangle for solution of the Fermats Last Theorem. so lets suppose that the sollution exist, a^n + b^n = c^n so if we sqare the equation it will hold true that (a^n + b^n)^2 = (c^n)^2 so -----> …

Hi, I am looking to find a componentwise combination operator to resolve a problem. I would really appreciate help (maybe this sounds obvious to you, but combinatorics are not my speciality). I want to operationalize a problem in which I would repeat a componentwise operation for all possible pairs from a given set, but where the ordering of pairs does not matter. For instance, let A be the set \{1,2,3\}. I would need an operator that means something like A <<insert operator here>> A = \{(1,2),(1,3),(2,3)\}. This is close to be a Cartesian product, but the Cartesian product A X A gives \{(1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3)\}, …

I dont understand this at all, i dont seem to get the logic in this, ive seen my helpful little rules to memorize but i dont seem to recognize it when i face a factoring problem, is there any tricks or some logical concept on this topic?

I am having trouble understanding the translation of the definition of subclasses in the text I am currently studying which covers First Order Predicate Calculus /w Equality. The definition (inclusion) [math] A \subseteq B \leftrightarrow \forall z \left( z \in A \rightarrow z \in B \right ) [/math] is being translated as "A is a subclass of B, or a subset of B if A is a set, or B includes A" in the text. The definition (proper inclusion) [math] A \subset B \leftrightarrow A \subseteq B \land \exists z \left( z \in B \land z \notin A \right ) [/math] is being translated as "A is a proper subclass of B, or a proper subset of B if A is a set, o…

My mom said i wont be able to do maths:confused: guess what i looked at it i laughed:D and i did it . My motto for those who struggle with maths ''we see it we do it'':cool:

x=1 y=25% x=2 y=40% x=3 y=50% x=4 y=57% x=5 y=62% x=6 y=66.60% this is based upon how 1/4=25 2/5=40 each time the numerator and denominator both increase by one. but I can't figure out the equation!

I'm not sure if this is in the right sub-forums. If not, please move it. If I were given two points (x,y) and (x',y'), how can I determine the angle they form? Let's assume that (x,y) is the vertex of the angle, and the other line the angle is to be measured from is (x+1,y). If the angle is to be defined by the area swept from (x+1,y), pivoting around (x,y) is a counterclockwise direction, and reaching the line connecting (x,y) and (x',y'), how would one calculate the angle?

my school allows you to skip a class if you can pass the finial exam so what is the best way to prepare for an algebra1 exam when you haven't taken the class?

hello, so i'm currently working on a linear programme optimisation, and i was wondering if someone could help me with a bit of maths. and i need to represent abs(x) in terms of a linear equation (so no root(x^2) allowed) everyone knows what absolute(x) or |x| represents and it's pretty simple to do in your head, but I was wondering if there were any computational methods or equations for evaluating this? also i'm looking specifically for a linear method so i cant have if statements and such also are there any functions that result in x = x if x>=0, x = 0 if x<0 if so what is it called? any help is appreciated, thanks

try to think of the Screen of a Computer with resolution of N x M in pixel .. as a Matrix of size n x m so i was thinking about a way to work on a matrix using iterators, and a recursive factorization function for the diagonal ... if we use a pattern/series/stream iterators, but we want to make a unique representation but what if we use mathematics to solve this problem, and exactly i mean Linear Algebra try to think of the screen as n x m Matrix, then we can have n horizontal iterators + m vertical iterators, and the intersection of different iterators give us the ability to control the pattern, .. also, we can try using diagonal-point recurs…

The Kronecker product of an argument X and a 2x2 matrix, increases the dimensions of each argument X individually. If each argument X is a scalar value, it now becomes a 2x2 matrix. How are these arguments now aligned with each other and the other elements in the resultant matrix? For example: In a matrix T, if we have several values and a certain number of 0s like: T= [0 1 0] [0.3 0 0.7] [0.1 0 0] For the new matrix TT, we perform the Kronecker product of the values in the second diagonal (i.e. of 1 and of 0.7) with a 2x2 matrix Q which is Q is [0.8 0.2] [0.4 0.6] and the Kronecker…

So i just got back from a Group Theory exam. One question's bugging me. |G|= 6655. Prove that G has nontrivial characteristic subgroups H & K, K a proper subgroup of H. Now i've proved that there exists a unique Sylow 11-subgroup (you guys may confidently assume this to be the case if you don't want to do it yourselves), thus it is normal in G. Let's call this subgroup H. We proceed to show that it is a characteristic subgroup of G. If f is any automorphism in G, the order of H is the same as that of the order of H under f (Hf is a subgroup of G). Since H is the unique Sylow 11-subgroup, H=Hf (if not, because Hf qualifies as a Sylow 11-subgroup of…

i keep seeing this problem and i dont see how it posible

In the following equation, [math]\frac{x(a)}{2}[/math] = [math]\frac{x(b)}{3}[/math]= [math]\frac{x©}{4}[/math], what is the smallest possible value of: [math]\frac{a+b+c+x}{2}[/math] In my case, I got 9.5. Are there any lower answers?

Hi! If [math]H<G[/math] ,[math]K<G[/math] and [math]K\subseteq H \subseteq G[/math] then ¿ [math]K<H[/math] ? Thanks!

Can we prove : if (1/a)>0 ,then a>0 ??

When defining the dot product, the cosine of the angle between two vectors is chosen. Why not the sine? What advantages are there by choosing the cosine?

for the following problem: For all ,a : if [math]0\leq a\leq 1[/math], then [math]\sqrt{1+\sqrt{1-a^2}} +\sqrt{1-\sqrt{1-a^2}} =\sqrt{2+2a}[/math].....................................................................................A the following proof is suggested: [math]0\leq a\leq 1\Longrightarrow 0\leq 1-a^2\leq 1\Longleftrightarrow 0\leq \sqrt{1-a^2}\leq 1[/math][math]\Longrightarrow(\sqrt{1+\sqrt{1-a^2}}\geq 0\wedge \sqrt{1-\sqrt{1-a^2}}\geq 0)\Longrightarrow[/math] [math](\sqrt{1+\sqrt{1-a^2}}+\sqrt{1-\sqrt{1-a^2}})^2[/math] = [math]1+\sqrt{1-a^2}+1-\sqrt{1-a^2}+2\sqrt{(1+\sqrt{1-a^2})(1-\sqrt{1-a^2})}=2+2|a|=2+2a[/math] And since [math] 2+2a\geq 0[…