Linear Algebra and Group Theory

Hi team, I'm really struggling with understanding how exponents make sense. According to a basic google definition: "An exponent refers to the number of times a number is multiplied by itself." This would indicate to me that 2 to the power of one would mean that 2 needs to be multiplied by itself... once. So 2 x 2 is 4. But I know that 2 to the power of 1 is actually one. Why is 2 squared not 2x2x2? I can only see two "x"s in there... Thanks for your advice.

When you're talking about volumes and hypervolumes, your set needs to be equipped with a measure function. The technicalities are too much to go into in a single post, but see the wiki link if you're interested. The standard measure used on the Reals is the Lebesgue measure where the length (in one dimension) is just the difference of the endpoints. So, if we're looking at the length of a point p, we need to just take p-p, which is, of course, 0.

/****************************************************************************** * Jouni Aro * * 20.2.20 * * * * The multidimensional algebra * * * * The possibilities of algebra is associated with a sort of tool, with which * * parallel causations can be harnessed and approximated into compact * * functions. When you draw a roughly smo…

Hello everyone, Why -2 .(-3) = 6 Why is it positive? Why we accept that (minus sign) times (minus sign) is positive ? What is its origin? Thanks in advance

Goldbach conjecture can be proved by using the Zermelo Fraenkel axioms in set theory. We apply the inductive method to prove that this conjecture is correct with all even numbers 2n with n ∈ IN. The process includes two steps 1. Prove that the conjecture is correct with n=2. 2. Assume that the conjecture is correct with a certain value n, prove that the conjecture is also correct with (n+1). We proceed with our process in sequence as below 1. With n = 2 => 2n= 4 = 2+2. The conjecture has been ascertained as correct in this case. 2. Now we assume that the conjecture is correct with n, the next step is we have to…

I had this idea and I think it belongs to Algebraic Topology but here it is: Imagine some object C in some non-Euclidean space K such that object C is constantly being transformed in some important way every time this object moves in K but only because it moves through K. If it moved through some space M, then it would transform but differently or not at all. Is there a theory about objects like these and spaces that manipulate objects.

Please help me the following equation, we need to calculate X relative to the other: \begin{equation} \tan{(2X-\phi_0)} = \frac{\rho\sin{\phi-a\sin{X}}}{\rho\cos{\phi}-a\cos{X}} \end{equation} Thank you very much!

Hello everyone, When you have square root of (-2)^10 which is the result? Because if I use the fraction notation results (-2)^(10/2) = (-2)^5 .... a negative number. On the other hand, if the power is calculated first the result is positive. Thanks in advance

Recently on a computational engine called wolframalpha, by accident I put this equation for a solution( I wanted a soln of another eqn). But this mistake is not so anymore, it showed me that almost by every mean mathematics can be manipulated. Anyways, the engine showed there exists no soln . for this. I tried this on few others and still got the same. I can't understand this. -1 and 1 both give 1 when multiplied by itself. No square of a number can be in negative form. So sqrt(x)=-1 should have the solution as 1. Can anyone explain this. Also, I searched this on certain sites and they explained with graphs of complex numbers containing p…

We consider a linear vector space V of dimension n. W is a proper subspace of V.We take a vector 'e' belonging to V-W and N vectors y_i belonging to W;i=1,2,3…N;N>>n the dimension of V. All y_i cannot obviously be independent the number N being greater than the dimension of V the parent vector space; k of the y_i vectors are considered to be linearly independent where k is the dimension of W. The rest of the y_i are linear combinations of these k, basic y_i vectors of W. We consider sums αi=e+yi;i=1,2…N (1) Now each alpha_i=e+y_i belongs to V-W. We prove it as follows If possible let alpha_i belong to W. We have e=αi−yi=αi+(−yi) …

I have this equation: See attached file I want to differentiate it with respect to X to get the minimum peak . Thanks for help

The transformation of a rank two covariant tensor has been considered. Then we proceed to consider a diagonal tensor[off diagonal components are zero:A^(mu nu)=0 for mu not equal to nu] to bring out a result that all tensors should be null tensors. A link to the google drive file has been provided.A file has also been attached considering the file attachment facility that has been provided by the forum. https://drive.google.com/file/d/10z63Xidgs3m8p04_C6ZiGh-8Q6KTwPsh/view?usp=sharing Incidentally I tried the Latex with the code button.But I am not getting the correct preview. Example \begin{equation}\bar{A}^{\mu\nu}=\frac {\partial \bar{x}^{\mu}}{\pa…

Since $\mathbb{R}$ is not first order definable, the statement $\forall x \in \mathbb{R}(x = x)$ is not a first order statement and thus not provable in ZFC. Does that mean we can assume $\exists x \in \mathbb{R}(x \neq x)$? If so, would this provide us with the basis for a field with one element?

Do most composite numbers have a large prime factor? First, I’ll define what I mean by a “large” prime factor. Let N be a number. If a prime factor of N is greater than the square root of N, then that factor is a large prime factor of N. As an example, 11 is a large prime factor of 22, because 11 is greater than the square root of 22, and so 22 has a large prime factor On the other hand, 3 is not a large prime factor of 12 because 3 is less than the square root of 12, and so 12 does not have a large prime factor. Below is a list of composite numbers with large prime factors: 6, 10, 14, 15, 20, 21, 22, 26, 28, 33, 34, 35, 38, 39, 42, 44, 4…

Let G be a group in which, (ab)^2 = (a^2)(b^2) for all a,b ∈G . Show that H = { g^2|g ∈G } is a normal subgroup of G.

Let A={ex,sin(x),excos(x),sin(x),cos(x)} and let V be the subspace of C(R) equal to span(A). Define T:V→V,f↦df/dx. How do I prove that T is a linear transformation? (I can do this with numbers but the trig is throwing me).

Question: Show that [math] {Ax = v_{0}}[/math] has no solution. I know [math] v_{0}[/math] is an eigenvector of A with eigenvalue 0, and the other eigenvectors do not have 0 eigenvalues. So, [math] {Av_0= \lambda_{0} v_{0}}[/math] [math] {Av_0= 0 v_{0}}[/math] [math] {Av_0= 0}[/math] So [math] {v_0}[/math] "is" the null space of A (since no other eigenvectors have eigenvalues of 0). So the question is asking me to prove there is no vector that when operated on by A gets to the null space. I can't think of how to prove this though, apart from saying "A operating on x can only give a vector that is 0 or in the column space"

How does one show the below matrix is a linear transformations I know I need to multiply something by (0,1) and (1,0)

I have been stuck on this problem for awhile and have no sense of it, I have gotten stuff written down but sadly don't have any confidence in my answer. Any help would be absolutely appreciated thank you!! This is what I have now. $$ \begin{array}{l} \mathrm{W}^{\perp}=\{p \in P 2 |\langle p, x+1\rangle=0\} \\ \langle p x+1\rangle=p(-1) x+1(-1)+p(0) x+1(0)+p(1) x+1(1)=p(-1) \\ (-1+1)+p(0)(0+1)+p(1)(1+1)=p(0)+2 p(1)=2 \mathrm{p}(1) \end{array} $$ since we are looking for polynomials such that $\mathrm{p}(0)=2 \mathrm{p}(1),$ and with the definition of $\mathrm{P}^1$ all polynomials $a x^2+b x+c$ such that $c=2(a+b+c),$ so the numbers a,b,c with…

Can someone help me? I'm trying to proof this: Proof det(AB)=0 where Amxn and Bnxm with m>n I created a generic matrix A and B, then I use Laplace transforms to conclude. I'd like to know if there is another way to proof that. Thanks!

Consider the Vector Space of Polynomial with degree <= 5. And the transformation to its second derivatives. Shows the transformation and its matrix em relate to the base of this space. Could anybody help me? Thanks!

I have read that the Tangent space and the Cotangent Spaces are Duals of each other. Why is this so? I can understand that both are vector spaces and so "qualify" on that account but are they uniquely qualified to be Duals of each other ? Is the fact that they have a vector in common (the point p on the surface) important*? Can the Tangent space be Dual to any other vector spaces or is the Cotangent Space the only possibility? *important in making them Dual Spaces

https://mathforums.com/threads/calculate-a-determinant-of-specific-a-matrix.348091/

Suppose I have a matrix P = \begin{bmatrix}x⊗x&x⊗y\\y⊗x&y⊗y\end{bmatrix} which is equal to another matrix Q = \begin{bmatrix}a&b\\c&d\end{bmatrix}. Is it possible to solve for x and y. Since xy and yx are equal to b and c and are not the same, the elements in the matrix P are outer product of two matrices x and y. Is it possible to solve for x and y such that xy and yx can be made predictable. Kindly let me know if I am not clear. Thank you for the kind help.

Hello! I need urgent help regarding the following question. Any help will be greatly appreciated. Thank you!