Linear Algebra and Group Theory

Otherwise known as the Laplacian. So I should apologize for the length of this post and for taking no prisoners here, but it would take me WEEKS to flesh out the background, so I dive in...... Given a finite [math]n[/math]-dimensional vector space [math]V_n[/math], define a space of [math]p[/math]-vectors by [math]\Lambda^p(V_n)[/math]. Now define the exterior derivative operator by [math]d:\Lambda^p(V_n) \to \Lambda^{p+1}(V_n)[/math]. It seems that to this operator one may assign an adjoint provided only one has an inner product on the space of [math]p[/math] -vectors and that these vectors, as differential forms, are exact. Fine. Thus, for [math]\al…

Hi, everyone! I'm stuck with this problem... Being [latex]T\in L(\mathbb{R}^n)[/latex] a linear operador defined by [latex]T(x_1, ... ,x_n )=(x_1+...+x_n,...,x_1+...+x_n )[/latex], find all eigenvalues and eigenvectors of T. By checking n=1,2,3,4 I guess the answer is: λ=n, x=(1,1,1) λ=0 (multiplicity n-1), x such as , [latex]\forall k \in \{1,...,(n-1)\}[/latex], [latex]x_k=1[/latex], [latex]x_n=-1[/latex] and [latex]x_i=0[/latex] in all other positions. For instance, for n=4, we have (1,0,0,-1), (0,1,0,-1), (0,0,1,-1). My problem is... how do I prove it for the general case? I'm trying induction, but I think I'm missing something... Thanks in advanc…

I am thoroughly ashamed to be asking such an elementary question, but WTF. I'll start at the beginning, for want of a better place to start...... Suppose that [math]S[/math] is a point set, and that [math]\mathcal{P}(S)[/math] is the powerset on [math]S[/math]. Then one defines a topology [math]\tau[/math] on [math]S[/math] by [math]\tau \subseteq \mathcal{P}(S)[/math], and ones calls the pair [math]S(\tau)[/math] as a topological space (though usually the parenthetical \tau is omitted in favour of the assertion that we are dealing with a top. space). Elements in [math]\tau[/math] are called the open sets in [math]S(\tau)[/math], and elements in the complement [ma…

I require some guidance on the correct terminology for the following: In a series of numbers... first / last members - ordinal / terminal (or other)? next / previous members - ordinal / adjacent (or other)? So far I have referred to them all as ordinal (as in 1st, 2nd, 3rd, ..., last), but I want to distinguish between first/last/next/previous. I only ask because I wish to properly name methods in some program code I'm developing. I am not a mathematician or statistician, etc, just a novice. Thanks for any help.

Is there a relation between the adjoint (conjugate transpose) and the classical adjoint (it is called other things) which is the matrix obtained by replacing each entry of a matrix by its cofactor and then one takes the transpose) Thank you

Consider a matrix A in Rn with eigenvectors vi and eigenvalues \lambdai Does anyone know of an efficient method to solve for the vector v := v1 + ... + vn in one go (rather than doing the whole spectral decomposition)? (I am especially interested in the case where A is real symmetric) Thanks p

How many numbers lie between 11 and 1111 which when divided by 9 leave a remainder of 6 and when divided by 21 leave a remainder of 12?

The reason I am writing to you is because yesterday I have sent the really amateur and wrong solution, sorry about that... I was really embaressed so I said I will try for the last time to crack the problem... 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 lets suppose a,b,c are coprimes Let us check the problem for odd powers of n. We can write now the equation (c^n + b^n) * (c^n - b^n) = (c^2n - b^2n) so that holds everytime, not specifically for the problem. ---------------> now the Fermats La…

Hi everyone, Given two different reference frames in a vector space; say left and right. v is a vector defined in the left frame and u is a vector defined in the right frame. What is the nature of a matrix A that can satisfy the equality u= A.v? Thank you

Hello. Sorry for my english. I have 3 coordinate systems [math]OI[/math], [math]OK[/math], [math]OE[/math]. System [math]OK[/math] defined in [math]OI[/math] by quaternion [math]A[/math]. System [math]OE[/math] defined in [math]OI[/math] by quaternion [math]B[/math]. I need to find quaternion [math]C[/math] that define rotation from [math]OE[/math] to [math]OK[/math]. My solution. 2 rotations [math]OI \to OK[/math] and [math]OI \to OE \to OK[/math] are equal. [math]A = B \circ C[/math], and [math]\tilde{B} \circ A = \tilde{B} \circ B \circ C = C[/math], i.e. [math]C = \tilde{B} \circ A[/math]. Is it right?

what is the difference between the following statements: a congruent (triple equal sign) to b mod (n) and a=b mod(n)?? do they differ in any properties ?

Was just playing around with the equations, when I found something strange. Suppose there is a number (1-x)^0.5, then pull out a -1 out of it so that it becomes i(x-1)^0.5 Now again, pull out a -1 from the root so that the term inside the root is reverted back to the original, So, it becomes (i^2)(1-x)^0.5 (1-x)^0.5 => -(1-x)^0.5 So, what does it mean? How is it possible to turn a positive real number to a negative real number without following any sort of symmetric algebraic algorithm?

How would I use Lie Algebras, observing infinitesimal transformations over a smooth manifold(non-complex Lie Group,) to gain insight into the geometric properties of some shape? What thoughts could I be observing when looking into the mathematics that this involves? I had looked at the concept of Lie Groups a number of years ago but abandoned it completely as it made no sense when I had. Just looking at the Wiki, most of it is pretty plain English now but Wiki leaves much to be said about proper application. I am beginning a study on topology and will be returning to group theory after I complete this study. I would like to have my thoughts in my mind focus…

how is summation of all phi(d) = n where d|n? thanks

Hi all, My question is: Is there a good source that defines "tensor rank" for me? Yes, I have seen several already, but perhaps I am mostly interested in a definition that is more mathematical (rigid)...For example, how do we define the rank of a multilinear form? If I have f: V x V x V...x V --------> k, what should the rank of this map be? Thank you

Simplify this Boolean expression please: F(a,b,c)=((c+a)' *(c⊕b)')' ⊕=XOR

i am a newbie to linear algebra My question is Let T of dimension mxn, be a linear map from A -> B. if n<m and rank of T is n, which basically means all of the vector space A is mapped,and null space is empty. Then why there is no inverse for T?? My understanding is that for A_inv to exist, all the vector space B should be one to one mapped back to A. This is not possible because vector space B is of high dimension(m) and thus have more elements than A, which is of low dimension(n). Is this view right??

Where [math] A [/math] is a class, [math] R [/math] a relation and x, y, z are all sets Is there another title for the concept of left and right-narrow? A search for left and right-narrow returns no relevant results other than the book that I'm reading and I wish to read other interpretations of the concepts that are along the same line; this being the concept of order and how left and right-narrow are relevant. I guess I'm missing the point on this one! I had skimmed over narrow rather quickly and had somehow taken it to be left and right restriction which it isn't. The development of order is not entirely intuitive as it is being presented in this book. …

I would like to express an SPD matrix A as: A = sI + B where B is also SPD, or alternatively as: A = I + B what constraints need to be imposed on B? Please let me know if this question is mot appropriate for this forum. thanks, Russ

Show that if A ε Mnxn is nonsingular and t ≠ 0, then tA is nonsingular and (tA)-1 = (1/t)A-1. I need to show an intense proof of this statement. Although I can grasp the concept in my head, I am unsure as to the mathematical reasons and theorems that prove this true.

for what value of parameter [math] m [/math] does [math] ||x^2-4x+3|-2|=m [/math] have 2-solution of different sign.

A moment ago, I posted something on the application of group theory to music in the thread real life applications of group theory. Here is an elaboration of that post, and another application of maths to music. Circle of fifths I mentioned in my previous post that the theory of intervals is based on the cyclic group of order 12. Here is another way of looking at musical intervals. It's a bit more complicated and less intuitive to the non-mathematical musician as it involves more mathematics – which suits people like me much better. Consider the set of all musical notes distinguished by their absolute pitch (or frequency measured in Hz). Define a relation ~ on th…

We know set calculation well. A={1, 2, 3} B={2, 3, 5} A cup B ={1, 2, 3, 5} A cap B ={ 2, 3} , etc. . New problem A={ red coat, red car, blue apple, blue dress} B={white coat, blue car, red apple, blue dress} A cap B = { blue dress} well known . New operation A (red)cup B= { red coat, red car, red apple} A (car)cup B= { red car, blue car} A (car, apple)cup B={red car, blue apple, blue car, red apple} How about this operation?

what happens to the phase of a sine wave when passed through a high pass filter circuit and a low pass filter circuit.does capacitor always produce a 90 degree phase shift for a sine wave at all frequencies.please tell me the response at different frequencies of signals.can we see the phase shift produced by an inductor to current in a cro , or only voltages can be seen in it? please comprehend.

for what value of [math] m [/math] does [math]x^2+(m+6)|x|+2m+9=0[/math] has 2-distinct real solution.