Linear Algebra and Group Theory

I saw a discussion that got me wondering about problems related to abstract algebra and group theory. How you formally use group theory to disprove 2=3?

With reference to Eq. 16 in the attachment, I need help in understanding how the working is done to obtain the answers in Eq. 17. Eqs. 10, 11, and 15 are the inputs needed to solve Eg. 16. I am puzzled how a matrix could be differentiated with respect to another matrix. I would highly appreciate it if an example calculation can be shown. Thanks in advance! Equations.pdf

I have a question regarding one example of eigenvector calculation. In Equation (1) of the attached example, should the first column of matrix A be written as 0.8x1 + 0.2x2 , rather than x1 + 0.2x2 ? Thanks in advance for your inputs. example.pdf

For the attached matrix equation above, can someone guide me on the steps to solve it? Highly appreciate any guidance ! matrix equation.pdf

Hello. If you have a matrix, a 2x2 say [latex] M = \left(\begin{array}{ll}a_{1} & a_{2}\\b_{1} & b_{2}\end{array}\right) [/latex] and supposing it is in some way expanded into a 3x3 matrix so that the third row or column is formed from some kind of multiplication of the first two, such as [latex] N = \left(\begin{array}{lll}a_{1} & a_{2} & a_{1}a_{2}\\b_{1} & b_{2} & b_{1}b_{2}\\c_{1} & c_{2} & c_{1}c_{2}\end{array}\right) [/latex] I was wondering if this process has a name because I am having trouble finding anything about it online. The reason for including a new row is because I am studying determinants and wanted to …

Hi, an easy one I guess but its been a while since I last stepped into a class room so I need some help. I need a function that fulfill f(f(a,b),c) = f(a, f(b,c)) e.g. simple multiplication or XOR, but in theory how does this family of functions are called? Thanks!

Hi I'm trying to solve this : assume: a*x^2+b*x+c>=0 for all x with a≠0 then have: b^2-4*a*c<=0 but I couldn't. I proved it when the quadratic equation is greater than zero a*x^2+b*x+c>0 for all x with a≠0 but not for greater then or equal to zero. So any one help in solving this please.

I understand that Minkowski (Poincare?) Space is a Group in Group Theory.(am I right so far?) Well I have (re) learned that for a Group to be a Group there are one or two (4 ?) basic preconditions and that these are (1) that the set must have a operator and (2) must also include an identity element , (3) be commutative in the operations and that (4) each element must have an inverse. Oh and (5) it must exhibit "closure" How do these conditions apply to Minkowski Space? Are the elements of the set spacetime vectors? Are all the vectors unit lengths or can they be any length? What is the operation ? Is the set infinite? Have I got the…

This is a question that doesn't have a solid question... Now we use math in everything we do... even on my laptop that I am using for this thread uses mathematics to solve and put in every single pixel on screen... We count words and letters in our language books and count the thousand habitable planets in our solar system. We draw graphs and charts for buisness and the economy needs to know how much they have of anything. But first the big picture. To define everything is like to define 'god'. It is impossble, because we all have a different oppinion or definiton of that. Science has something to do with math. We can't count the amount of atoms that exist in and …

There seems to be countless resources that describe how to draw a matrix, how to number the elements, how to add matrices, multiply them etc. However, there seems to be very little information describing what exactly the matrix represent and why it is important. Also, why are addition and multiplication defined as they are? Matrices always felt very non-intuitive to me and I hope by achieving better fundamental understanding I can overcome this. If you know of a resource that describes this well or would like to give your own explanation please do.

Hi, I know how to compute determinants and I'm familiar with the geometrical meaning of determinant as the scaling factor of a unit (point/square/cube/hypercube)'s area/volume by applying a linear transformation (using a matrix). However, I have several questions: Let's say I define determinant to have the above meaning. How can one derive the formula for computing determinant following just the visual/geometrical meaning? Let's say I have an arbitrary closed 2D polytope [latex]P[/latex] and I transform all of its vertices by a matrix [latex]\mathbf{A}[/latex]. Is [latex]\det\left(\mathbf{A}\right)[/latex] the scaling factor of polytope's [latex]P[/latex] area afte…

Hello. I would appreciate some help in determining how much sense the following really makes. I don't know very much about "writing mathematics" so any advice is very welcome. --- Let [latex]f:\mathbb{Z}T\rightarrow 0[/latex] be a periodic function with period [latex]T[/latex] and let [latex]g:\mathbb{Z}T\rightarrow \mathbb{R}[/latex] be invertible. Then the zeros of the composition [latex]f\circ g^{-1}:\mathbb{R}\rightarrow 0[/latex] is the set of all [latex]t[/latex] for which [latex]t=g\left(\frac{1}{2}Tz\right),z\in \mathbb{Z}[/latex], is satisfied. --- That is it. I can explain more if necessary. I welcome all constructive feedback, positive and negative. …

This is just so much fun. We will suppose that [math]U,V,W[/math] are vector spaces, and that the linear operators (aka transformations) [math]f:U \to V,\,g:V \to W[/math]. Then we know that the composition [math]g \cdot f: U \to W[/math] as shown here. (Remember we compose operators or functions reading right to left) Notice the rudimentary (but critical) fact, that this only makes sense because the codomain of [math]f[/math] is the domain of [math]g[/math] Now, it is a classical result from operator theory that the set of all operators [math]U \to V[/math] is a vector space (you can take my word for it, or try to argue it for yourself). Let's ca…

Hello everybody. I'm having a little bit of trouble understanding a passage of my textbook regarding a linear transformation and matrix multiplication, I wonder if you could help me out. So, I have this equation: [math] \dot x = \textbf{Fx} + \textbf{G}u [/math] Where F is some 3x3 matrix and x a 3x1 array. For now, these are the important variables. So, my objective is putting F in a specific format called control canonical form (A), which is: [math] A = \left| \begin{array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ a & b & c \\ \end{array} \right|.[/math] For that, the book shows a Linear Transformation in the variable x: [m…

Can the following system of equations be solved using Gaussian Elimination? [latex]\begin{bmatrix}s_{00} & s_{01} & s_{02} & s_{03}\\s_{10} & s_{11} & s_{12} & s_{13}\\ s_{20} & s_{21} & s_{22} & s_{23}\\ s_{30} & s_{31} & s_{32} & s_{33}\\\end{bmatrix}\begin{bmatrix}x^2_0 \\x^2_1 \\x^2_2 \\x^2_3 \end{bmatrix} = \begin{bmatrix} 1 \\ 1\\ 1\\ 1\end{bmatrix}[/latex] If one were to let [latex]w_i = x^2_i, 0 \leq i \leq 3,[/latex] then the above system is (trivially) transformed to [latex]\begin{bmatrix}s_{00} & s_{01} & s_{02} & s_{03}\\s_{10} & s_{11} & s_{12} & s_{13}\\ s_{20} & s_{2…

Hey, I started to learn what groups are... I asked myself, is the multiplication in R a group? My counterargument is, that there is no identity element, because 1*e = e*1 = e, expect for e=0. And the second one is, that there is no symmetric element, because: a*b = b * a = e, where b = 1/a and e = the no identity element. This works, except for a=0... Am I wrong in my argumentation, or is the multiplication in the real set no group? Thank you!

What are coefficients, really? The elementary answer is, for, say, a polynomial of the form y=ax^2+bx+c, that a and b are coefficients that are usually held to be constant or that might be construed as being variables in some cases, and that c is a constant that is, in the case of a polynomial, the y intercept. But what ARE coefficients? What is the formal definition of each one and/or what they are as a set? That is, if we say that the elements of a polynomial (or any function/mapping) are S={a1,a2,...,an}, what is this set by itself? And the answer should include polynomials but can include rational expressions or linear functions, though I know that in the line…

I am preparing myself for maths exam and I am really struggling with kernels. I have following six kernels and I need to prove that each of them is valid and derive feature map. 1) K(x,y) = g(x)g(y), g:R^d -> R With this one I know it is valid but I don't know how to prove it. Also is g(x) a correct feature map? 2) K(x,y) = x^T * D * y, D is diagonal matrix with no negative entries With this one I am also sure that it is valid but I have no idea how to prove it or derive feature map For the following four I don't know anything. 3) K(x,y) = x^T * y - (x^T * y)^2 4) K(x,y) = $\prod_{i=1}^{d} x_{i}y_{i}$∏di=1xiyi 5) cos(angle(x,x')) 6) min(x,x'), x,x' >=0 Please he…

Hello, I want to know if there is a formula to find the higher partial derivative of an inverse of a matrix with multi index. You find the formula in attach fil. I need this formula this weekend because I have a work to finich it and it rest for me just this one. Thank you

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I don't know what it's called but I'm trying to find a general formula a cosine function that's very similar to a regular cosine function except it's not symmetric about every local minimum, instead it's lopsided. It's like a sawtooth/ramp wave but it's not a triangle and it doesn't have vertical lines, it's just as if a sin(x) got squished on one side of every cycle. Like for any given cycle, it's really steep slope on one side, then after the maximu/minimum it's a really shallow and low slope until the next cycle. Once again this site is horribly glitched and it's not even letting me post links anymore, so I guess the only picture I can relate it to is a chi distributi…

Hello, My name is Mark, I am a math Major and Thomas Edison State College. I will be getting a B/S in Mathematics and an Mba in Real Estate. Possibly if there is time I will get a B/A in Philosophy. I am interested in the game of Casino Craps. There is a cult following out there that states that they can beat the game of craps and get a positive player edge with Influencing the dice the way they are thrown to eliminate or decrease the percentage of the 7 ratio. There is another school of thought that energy and vibe some how control and can at times manipulate a random game of chance. Since Dice have no memory unlike Card Counting in Blackjack. Craps is a negative expecta…

1)prove that the reduced row echelon form of an n by n matrix either is the identity matrix or contains at least one row of zeros 2)let the homogenius system Ax=0 whose augmented matrix is A/O be equivalent to R/0 where R is an equivalent matrix in reduced row echelon form.Let there be n unknowns and r non zero rows in R If r is smaller than n then the linear system A/O has infinite number of solutions. prove it

I'm having a horrendous amount of trouble understanding Vector Matricies. I think what really stumps me is this: If I have a 3x3 vector like this: 1 0 0 0 1 0 0 0 1 Or to apply a value to each I would put: (which is probably technically wrong) a d g b e h c f h I understand if I have a vector representing a shape on a Cartesian plane and I want to translate, rotate, refect or something else it.... : The vector could look like this: (I'm not sure what the 3 would represent) 5 2 3 ......I can multiply or add the 3x3 matrix to it and by changing the values a to h I can effectively translate, rotate, reflect or the matrix to a mor…

Real Pythagorean triples are numbers like 3, 4, and 5. If a, b, and c are three positive real numbers and [math]a^{2}+b^{2}=c^{2}[/math], then a, b, and c can be used as the sides of a right triangle. So real Pythagorean triples have geometric meaning. But what if a, b, and c are not real numbers? For example: [math](-13+6i)^{2}+(6+22i)^{2}=(3+18i)^{2}[/math] If three complex numbers form a Pythagorean triple (and they are not real numbers) do they have a geometric meaning?