Linear Algebra and Group Theory

k0∑xj0 + k1∑xj1 + k2∑xj2 +,…,+ kn∑xjn+0 = ∑yjxj0 k0∑xj1 + k1∑xj2 + k2∑xj3 +,…,+ kn∑xjn+1 = ∑yjxj1 k0∑xj2 + k1∑xj3 + k2∑xj4 +,…,+ kn∑xjn+2 = ∑yjxj2 . . . k0∑xjn + k1∑xjn+1 + k2∑xjn+2 +,…,+ kn∑xjn+n = ∑yjxjn One of the most beautiful algebra formulas is the least squares polynomial formula.

A definition of an object \(x\) is called impredicative if the expression used to explain the value of \(x\) contains mention of \(x\) itself. The example that I want to ask about is perhaps illustrative enough: In Group Theory, \(e\) is a neutral element for \(G\) if \(ex = xe = x\) holds for every element \(x\) of \(G\). Since \(e\) is itself an element of \(G\), this is a typical example of an impredicative definition. This statement is the axiom that a group has a neutral element. Therefore I began wondering if perhaps Group Theory is an impredicative theory. My original motivation comes from a nice booklet written by Edward Nelson titled Predicative Arithm…

Are there any good images that can be used as a hook for an elementary number theory course?

By eliminating symbolic signs for multiplication, complex spaces are possible. I studied this problem for over a decade until it was resolved. Complex spaces fulfill all municipal conditions. Please find the code below to test the municipal rules. ! Moderator Note No one should have to reverse engineer your code to understand what you are talking about. Explain what you want to say.

Hi, evevyone I met one question, when I read one book about linear algebra.There is one matrix A which is m by n matrix, and it has pseudoinverse A+, how can I prove the A++ = A? Could some help me to give the detailed proofs? Many thanks in advanced. Best regards, Ning

Please help check my answers, it is not cut and paste, I wrote it through latex and I used a snipping tool to cut and paste it

Hlw, can anyone check to see if I'm right

20x - 6(0.6x - 2) - 8(4 + 0.8x) = 0 Answer is 2 but I got 0.54 Nevermind I got it right, I just messed up decimal and multiplication

How many exist subgroups F in G such that F=(C8)^5 and H∩F is C8? I yet understood that there are 2⋅8^(i−1)−4^(i−1) ways to choose C8 from (C8)^i. But I don't know what I can do next.

what are the applications of Langrange theorem in daily life? where we use it?

Hi there, I'm having trouble understanding a step in a textbook I'm currently reading (MIT Lectures on Dynamic Systems and Control); I just took a picture of the text to save myself the trouble of copying it over. W is a unitary matrix, [math]\sigma_i[/math] are the singular values of A. I haven't the slightest clue as to why this might be, and seeing as I've search the internet and found no refference to this and the author's don't elaborate I assume it's fairly simple and should be well within my reach, but for some reason I just don't get it. Thanks in advance for any help.

Given a number n, what is the best approach to find the most compact representation for n in terms of sums of powers, such that the bases can't surpass a given value? As in this image, the betas are the base and are limited in range (they are positive and can't surpass a given value, say, for instance, 2^32). Amongst all possible representations, how could we find the one(s) that have the less non-zero betas possible? As an example of an instance of the problem, suppose n = 558545864083284009. Amongst all possible such representations of this number, there is at least one that requires only two betas: beta1 = 2, beta21 = 7, such that 2^1 + 7^21 = 55854…

I am writing an introduction to a first course in elementary number theory. The topics are linear Diophantine equations, modular arithmetic including FLT and Euler's Generalization, quadratic residues and Non - linear Diophantine equations. How can I write an introduction to this showing linkage between the various topics and hook potential students to do this course? What is the motivation on covering these topics?

I may delete post if equations do not render in LateX, standby, preview doesn't show anything. Greetings Science Forums and Mathematics Community! I am Edward Solomon, and in my research into prime numbers and the Andrica Conjecture I geometrically derived a formula concerning the interior sum of partitions inside a rectangular rank-n tenor, which coincidentally made a striking statement equivalent to Fermat's Last Theorem. Here is the formula: Let c be an integer. Let a be an integer, such that a < c. Let k = c - a. \[ 0 = \sum^{j=k-2}_{j = 0} (-1)^j\binom{k-2}{j}(c-2-j)^n \] From the above equation it follows that: \[(c-k)^n …

here is some about euclidean division. The article states that although euclid doesn't know about the theorem but still it named after him (see) . The reason for it is also mentioned (written as "The term Euclidean division was introduced.........") but i can't understood it (probably because i'm not familiar with abstract algebra). Can someone help me understanding that.

Lately, I came up with a problem that I cannot solve and seems to be paradoxical. The problem is simple: A = x0 + x1 + x2 + x3 + x4 +... with x > 1. The sequence goes on to infinity. If we multiply A to x, we will get A.x = x( x0 + x1 + x2 + x3 + x4 +...) A.x = x1 + x2 + x3 + x4 +... A.x = A - x0 = A - 1 A- A.x = 1 So : A = 1/(1 - x) This clearly cannot be true since if x is 2, A would be -1 instead of infinity. However, I don't know what did I do wrong or does the problem even make sense or not. I would be thankful to receive help from you guys. Thank you.

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Axiomatic set theory ZFC is inconsistent thus mathematics ends in contradiction LINK DELETED Axiomatic set theory ZFC was in part developed to rid mathematics of its paradoxes such as Russell's paradox The axiom in ZFC developed to do that, ad hoc,is the axiom of separation Now Russell's paradox is a famous example of an impredicative construction, namely the set of all sets which do not contain themselves The axiom of separation is used to outlaw/block/ban impredicative statements like Russells paradox but this axiom of separation is itself impredicative …

Hello everyone, I originally intended to write an essay on Quantum Geometry for an undergraduate Geometry class, but I ran into some problems. Although there is plenty of information available on thebasics of the Heisenberg Uncertainty Principle and Planck's Constant, there doesn't seem to be any information available on how this affects the behavior of points, lines, plains, and half planes (the undefined terms of neutral geometry) on a quantum scale. Would it even be possible to use the basic theorems of Neutral Geometry? Does a Euclidean model fit Quantum Geometry? Is a completely different set of axioms required to work with Quantum Geometry? Do I need a good und…

I understand the rank of a matrix to be the maximum number of linearly independent rows (or equivalently columns) in a matrix. The spark seems to be the minimum number of columns which are linearly dependent. I'm missing some subtlety because these seem to me to be the same thing. Does anyone know any good resources on the distinction, i could only find the wikipedia page and that is a bit sparse.

I am studying a Kronecker product to solve a task in signal processing. it is a full correlation matrix of the channel. I know that my question is: How to find R_t , R_r? $ I can find a solution according to Wikipedia article about that ( operation of the Kronecker product). The answers are a unit matrix (2x2) and 0,8*matrix of ones (3x3). But I would like to understand a mathematic calculation. This matrix is easy for calculation without knowing about it. I hope that someone can provide me with more information about the calculation or can explain to me how to find the two matrices of the Kronecker product

I need really your help. I am trying to understand block diagonalization algorithm for multi-user MIMO system. But I guess My main problem is I don't understand the mathematics apparat of calculation SVD and other stuff. let's assume that the channel matrix looks like: Now I would like to calculate HP, where P is a projection matrix into the null space of H. Question 1: What means? **My suggestion:** = 1/2 (1 1 1 1), it is the first row V^H Question 2: Can someone explain me this PS please provide me the exaples

1)By assuming only two possible values for its unit content, Boolean algebra allowed a series of associations for its comprehension.Given the truth table below,X Y?0 0 01 0 00 1 01 1 1The result that produces these values, which is symbolized by a query will be:Alternatives:The)X + YB)X. YW)X - Y(d)X + NOT Yand)X. Y + Y2)By assuming only two possible values for its unit content, Boolean algebra allowed a series of associations for its comprehension.Given the truth table below,X Y?0 0 01 0 10 1 11 1 1The result that produces these values, which is symbolized by a query will be:Alternatives:The)X + YB)X. YW)X-Y(d)X + NOT Yand)X. Y + X3)By assuming only two possible values f…

Is it possible to compute the inverse of a 4*4, 5*5 .... Matrix ? If yes, will it be difficult computation ? If no, why ? Thanks & Regards, Prashant S Akerkar