Linear Algebra and Group Theory

Hello, I need some help understanding the concept of an arrow as a vector tensor. A [a,b] tensor can be thought an arrow starting in (0,0) and ending in (a,b). Can "arrows" begin somewhere else than (0,0)? If so, how does its "1 x n" notation look like?

I have found what I think is a simple, possibly efficient, algorithm for a Prime Number Sieve. This sieve is a process of taking a value from the set { 6x-1 U 6X+1 }, (5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37 ... N ). Using that value as a starting point to move through the same of numbers, based a simple pattern that I have found, to mark / eliminate the non-primes. Repeating the process for all the primes. The process using the multiple of 6, a pattern of the odd numbers, & an alternating switch to identify members of the set to be marked as non-prime. The steps of this process are Goto the starting value Inside Loop Add to the multiple the First Constant t…

Here is the problem: A square plate has vertices at A(1,1), B(-1,1),C(-1,3), and D(1,3). It is translated by 5 units along the +x-direction and then rotated by 45 degrees about P(0,2). Determine the final coordinates of the plate vertices. My answer: I will take point A(1,1) and transform it only. Order of transformations: First translate then translate+rotate+translate-back Initially, This is the translation matrix This is the A's coordinates matrix. Now we multiply them and get For rotation, we translate first by -2 in y-axis. we still get the same result of matrix. then rotate by 45 degrees After translating back to by +2 in y, we still get the same as above. The r…

So, while studying the Characteristic Roots and Characteristic Equation of a Matrix, I came across 'Spectral Decomposition of a Symmetric Matrix' which I didn't fully understood. Please help me in this and suggest me a source for the same because apparently it is not given my text books and my professor just mentioned this in the lecture.

Consider the following. The Greeks are credited with advancing geometry by large amounts, but consider if one could go back in time and teach Algebra to the ancient Greeks. How would that knowledge hurt their development of geometry and how would it have helped further the advancement of algebra. How would our world be different today?

Suppose \(V, W\) are vector spaces overthe field \(\mathbb{F}\) and that \(L:V\to W\) is a linear transformation. We know that the dual spaces \(V^*, W^*\) exist. What would be the dual transformation i.e. \(V^* \to W^*\)? I'm tempted to suggest it has something to do with pullbacks, but I can't seem tp get it to work

Show that if f(x) is an irreducible polynomial over K and K ⊂ L such that the degree of L over K and the degree of f(x) are relatively prime, then f(x) is irreducible over L. Let x0 be a root of f(x) and n be a degree of f(x). Since f(x) is irreducible over K, the degree of K(x0) over K is n. Let r be the degree of L over K. We know that n and r are relatively prime. Let's assume that f(x) is reducible over L. Then x0 is root of one of its factorizing polynomials. Call its degree, m, where m < n. Since this polynomial is irreducible over L, the degree of L(x0) over L is m. Since L includes K, L(x0) includes K(x0). Call the degree of L(x0) over K(x0)…

Make a list of all irreducible polynomials of degrees 1 to 5 over the field (0, 1). In the order of their degrees, this is my list: x, x+1; x2+x+1; x3+x+1, x3+x2+1; x4+x+1, x4+x2+1, x4+x3+1, x4+x3+x2+x+1; x5+x2+1, x5+x3+1, x5+x4+x2+x+1. Did I miss any? Is any of the above reducible?

I'd like to make textbook exercises, some as a refresher and others new to me, and I hope that mathematicians here will take a look and will point out when I miss something. Here is a first bunch. Which of the following subsets of \(\mathbb C\) are fields with respect to the usual addition and multiplication of numbers: (a) \(\mathbb Z\)? Not a field. E.g., no inverse of 2. (b) \(\{0,1\}\)? Not a field. Not even a group. (c) \(\{0\}\)? Not a field. No multiplicative identity. (d) \(\{a+b\sqrt 2; a, b \in \mathbb Q\}\)? Yes. (e) \(\{a+b\sqrt[3] 2; a, b \in \mathbb Q\}\)? Not a field. Can't get inverse of \(\sqrt[3] 2\)…

Consider the following proof: My question is, does it in fact use induction? It says, "Assume now that the theorem is true for k-1 elements," but I don't see this assumption being used in the proof to advance from k-1 to k elements, which would be an induction step. Actually, this assumption is not used at all, AFAICS.

hi everyone,im a 17 year old ,studying pcm in india and i am a bit interested in the subjects that i learn. so a few days ago i heard about smth called perfect numbers. so my question is that why do not we have a prime perfect number?

Hello, An overdetermined system of linear equation y = A x + z with y vector of known real numbers of dimension m; x vector of unknown real numbers of dimension n; z vector of Gaussian noise of dimension m and A the known coefficient matrix. it is characterized by 3 aspects: 1) The unknown x exhibits elements with order of magnitude difference among them. example: x is 4 elements and I know in advance that two of them will be around 10^4 and 2 around 10^0 2) The vector z is a noise and each of its element is a Gaussian number with zero mean and known variance. Bas…

Hello, In algebra and number theory, it can be demonstrated that 0^0 = 1 is true, but in analysis, 0^0 is an indeterminate form. Through the calculation of limits, it can be equal to 1, to other finite values, diverge, or even not exist. This is why, when faced with this issue, mathematicians have conventionally set 0^0 = 1. So, let's be bold: instead of saying that 0^0 does not have the same value in all contexts and that 0^0=1 is a convention, let's change the notion of the number 1 so that 0^0 = 1 in all contexts. In this new conception of the number 1, it would be both a number and an indeterminate form, meaning that it could be equal to 1, to ot…

Hello my old community. I'm going to be publishing my proof soon, but I've solved the conjecture about 7-8 years ago. Ive been sitting on the solution for a long while. Since the introduction of LLM's ive taken the opportunity to test the consistency of my answer. Following the solution, ive found a couple interesting patterns in primes reciently as well. It seems to have a connection to the reinmann hypothesis. Namely the 1/2 part. - The density of primes is contained in the first half of all numbers. For m, m composite, m-1 =2n. The primes that constitute 2n are all from 0 to n and none from n+1 to 2n. Meaning the relative den…

We have: P(A|B) = P(B|A)P(A)/P(B). Is P(B) = P(B|nothing) or is P(B)=P(B|everything)?

My new issue in my journey to try to understand infinity concerns the "ends" of infinity. I was told on here that the infinite sum of 1/2^n = 1, and not just gets close but actually equals 1. I can't help but notice that we are giving infinity a definite beginning point at 1/2 and a definite end point at 1. What could n possible equal to get to this point? If this last point really is a solution to the equation, then wouldn't it have to be 1/infinity, or in other words, the "infinity-ith" point? If so, how can it be said that the natural numbers can numerate all points of a set of size aleph-null?

Ann and Bill independently work on examples of Cantor's diagonal argument. Ann: 111001... 000111... 101100... 110011... 000110... 010110... transforms diagonal D1 to alternating '01' sequence T1, 010101... which can't appear in any list per the cda. Bill: 000110... 010110... 110001... 000111... 101100... 110011... transforms diagonal D2 to alternating '10' sequence T2, 101010... which can't appear in any list per the cda. If D1 and D2 appear in any list, they must be members of the complete list. T1=D2. I.e. a missing sequence is only relative to an individua…

This is about the paper "Fast contact force computation for non-penetrating rigid bodies". Has anybody made sense of this? I have questions. Do I completely neglect the unprocessed a's that are neither mc nor c? If the sign for an unprocessed "a" changes, should I stop and pivot and put that index in either nc or c? I know that the unprocessed "f's" remain zero, but what if the sign of an unprocessed "a" changes? Do I actually perform a pivot operation on a matrix? Or is that done by moving between a "c" and an "mc"?

Scale over 20 years ago. I program now the fractal carbon colors. Each pixel contains 256 * 256 mosaic information, resulting in an image resolution of 216 (65 536).

What is basic algebraic methods and is it part of calculus? or part of precalculus?

The two sets N (naturals) and I (integers) have a one-to-one correspondence and are said to have equal size/cardinality. But if we put them one-to-one in a specific way, such as the naturals to the naturals from I, we see that the naturals of I get used up leaving 0 and the negative integers. This seems to show that a correspondence from N to I can also not be one-to-one. The curiosity I get from this is just too much. It almost seems like this is an example of something that can be proved to be true and can be proved to be false. I would have to think that my problem is that I am not allowed to correspond the naturals to only the naturals of the…

What is functions simplified and detailed?

Is a quadratic equation a part of algebra?

I have a question about something similar to the twin prime conjecture. If you have two consecutive numbers, is there a limit to how smooth they can be? Here is the basic definition: If the largest prime factor of a number is less than or equal to the nth root of that number, then the number is nth root smooth. I am curious about pairs of consecutive numbers that are both nth root smooth for large values of n. Let’s call them twin nth root smooth numbers. For example, 2400 and 2401 are consecutive numbers. The largest prime factor of 2400 is 5, and 5 is less than the 4th root of 2400, so 2400 is 4th root smooth. The large…

I need to devise a module for next academic year which is an introduction to pure mathematics. They need to use this module as a step stone module such as number theory, group theory, combinatorics, and real analysis. What should I cover to make this interesting and be used as a hook for them to be motivated to do pure mathematics? Should have an impact on them. It will be a first-year university undergraduate module.