Linear Algebra and Group Theory

Galois's theorem describes the relation between subfields and Galois groups on the field. I tried to extend it to the relation between subgroups and automorphisms on the group in the following site; http://hecoaustralia.fortunecity.com/group/galois.htm

Hi I'm having trouble reducing these matrices: 1 5 a -4 -5a -16 a 20 16 and k 3 4 3 5 8 3 -2 k I can get them both down to 2 zeros in the first column, but can't seem to reduce to one in the second column. Thanks for any help!

A group is often represented with matrix, however when the group is generally expressed with <S;R>, for example <{x,y};{xyxy^(-1)}>, it sometimes requires hard work to find appropriate representation used with matrices. I tried to represent the group expressed with <S;R> by affine transformation in the following site; ttp://hecoaustralia.fortunecity.com/affine/affinerepresentation.htm It can induce the representation of the group automatically by the calculation based on <S;R>. >Would appreciate it if you shared some of these ideas on the forum instead >of linking to ad-riddled pages! Any (constructive) questions welcome! I'll…

I did the work but not sure if its right, also my professor likes us to include every detail(including all the Vector space Axion) , if there is another way of proving it, more elegant,, please help,, thanks 1) Let V be the set of all pairs (x,y) of real numbers with the addition + and scalar multiplication* defined by: (x1,y1)+(x2,y2)= (x1 + x2 , y1+y2) and c*(x,y)=(x,cy) Show that V with the above operation is not a vector space. Find at least one axiom that fails and give an example showing that the axiom fails.. ***Let α = (x, y) Then for real numbers a and b we have (a + b) α = (a + b) (x, y) = ( x, (a+b)y ) Now aα = a(x, y) …

Cauchy's equation is expressed as, f(x+y) = f(x)+f(y) In the following site: http://hecoaustralia.fortunecity.com/cauchy/cauchyequation.htm I tried to expand the Cauchy's equation to general group homomorphism.

Hi, I haven't been alb eot figure out how to figure out how to do the coefficient of determination on the TI-86, I am able to do it on the TI-84, but not the 86. Anyone have any ideas? I have searched on google and there were keystrokes on how to do it on the 86 but while following the steps, some of the buttons they wanted me to press didn't exist on the 86. So I have no idea now. Thanks.

Verify that H: R^3--->R^3 is a bilinear form ........................................... H[(a1,a2,a3), (b1,b2,b3)] = a1b1 - 2a1b2+a2b1- a3b3 in my book (a1,a2,a3) and other vector written in column form.

If anyone could explain how the following is done, it would be greatly appreciated! Consider the set L of all linear transformations. Let L_1: V -> W and L_2: V -> W be linear transformations. Define a vector addition on linear transformations as (L_1 + L_2): V -> W, where (L_1 + L_2)(v) = L_1(v) + L_2(v). Define also a scalar multiplication on linear transformations as (c * L_1): V -> W, where (c * L_1)(v) = cL_1(v). Using these operations, we may consider L a vector space. Here is the question: Show that for every L_1 ε L, there exists some L_2 ε L such that L_1 + L_2 + 0_L.

I was wondering if someone could explain to me the easiest method for determining if two functions are inverses of eachother? Thanks

It's a Basic Question and therefore should be asked if other things are built upon it. why is a negative times a negative a positive and a negative times a positive a negative and similarily for division? If i missed something basic back in grade 3 please be gentle thanks, A Fool

I'm not really sure how to approach the following: Let A be a commutative ring for which aA=A for all a not equal to 0. prove A has no zero divisors. any ideas?

Let me preface this by saying, I am not a professional mathematician, and I do not know that much about Lie Algebras. In this question I will be using the strong Einstein summation convention. The summation over the greek indices will be over numbers 0,1,2,3. My question involves comparing and contrasting the Lie Algebras of the Poincare group versus that of the group [math] GL(4,\mathbb{R}) [/math] with translations [math] \mathbb{R}^4 [/math]. In special relativity coordinate transformations are given by [math] {\mathbf{\bar x}} = \Lambda {\mathbf{x}} + {\mathbf{a}} [/math] where [math] \eta = \Lambda ^T \eta \Lambda [/math] and [math] \eta _{…

can any one of u tell me if a transfer function has two poles say fp1 and fp2 and if fp1 is very smaller than fp2 say fp2=4fp1 then how the 3db frequency is less than 6% smaller than fp1 plz reply any one of u coz i m having great confusion

I never understood this concept. What proof backs this up? This argument is sometimes also presented as "2 + 2 = 5 for large values of two." This doesn't make sense to me whatsoever. Can someone please help?

I'm working on a project to show linear algebras applications in electrical circuits. I don't have alot of experience with circuits, and I don't have any experience with differential equations, so I'm not quite sure how slim my options are. Anyways, I've found out how to use augmented matrices to find loop currents, and now I've hit a wall so to speak. I've googled for ideas online but some are either too intense or not applicable. We're just starting eigenvectors, so I barely know how to work those. But if you can think of anything that I could expand on in my project given my circumstances, I would appreciate hearing them, thanks.

how we can calculate the order of group that i know, but Question is this A abliean group of order 28 (a) has exactly two subgrop of oredr 3 (b) no subgroup of order 3 © more than two subgroup of oredr 3 (d) one sub group of order 3 what will be the correct answer among these? Thanks for answer..................

The following problem is puzzling me so any help would be appreciated! Consider the linear transformation L: M_22 -> R^4 defined by L of (a b c d) = (-10a - c + 2d, 5a + b - d, 2c + d, 9a + 17b + 2d) Compute ker(L). Okay, so far I've put the matrix: 10 0 -1 2 5 1 0 -1 0 0 2 1 9 17 0 2 into row reduced echelon form to get 1 0 0 -1/4 0 1 0 1/4 0 0 1 1/2 0 0 0 0 And from this, I've gotten that a = 1/4 d b = -1/4 d c = -1/2 d and d is arbitrary But how do I notate ker(L)?

Consider the quadrilateral (namely Q) in R^3 formed by the points (1, 0, 0), (2, 0, 0), (1, 1, 3), and (2, 1, 3). (a) What should the coordinates be for the figure R we get by rotating Q counterclockwise in the x-y plane by 45 degrees, then dilating it by a factor of 3/2, then translating it along the vector (-2, 1, -1)? Okay, so what I did was I used the matrix cos45 -sin45 0 sin45 cos45 0 0 0 1 After this, multiplied the identity matrix for R^3 by 3/2 and then multiplied it by the matrix with 45 substituted for theta. Then T(x,y,z)=(-2+3sqrt(2)x/4-3sqrt(2)y/4,1+3sqrt(2)x/4+3sqrt(2)y/4,-1+3z/2) I substituted each of the quadrilateral poin…

If I have z= ||v||u + ||u||v, can I say that (z/uv) = (||v||/v) + (||u||/u) ?

How would you find the dimension of the subspace of [math]\mathbb{P}_{3}[/math] spanned by the subset [math]\{t,t-1,t^{2}+1\}[/math]? How would you find a basis and dimension of the given subspace [math]\mathbb{R}^{3}\}[/math] [math]\{[a,a-b,2a+3b]|a,b, \in \mathbb{R}\}[/math] as a subspace of [math]\mathbb{R}^{3}[/math]?

So I am working through Hoffman and Kunze, and in the chapter on canonical forms I am having some difficulty due to the lack of examples. Using another book, I figured out how to calculate the Rational and Jordan canonical forms for a given matrix (linear operator). And I assume that the so called "rational decomposition" has something to do with the Rational Canonical Form? But what exactly is the relationship? For a finite dimensional vector space V and linear operator T, There exists r non-zero vectors in V and r respective T-Annihilators so that the direct sum of the cyclic subspace generated by the vectors with respect to T equals V AND the T Annihilator for eac…

Please any idea on this one,Given the symmetric matrix [math] A=\begin{bmatrix}0&2&2\\2&0&2\\2&2&0\end{bmatrix} [/math] Find an orthogonal matrix P so that [math] PAP^{-1} [/math] is a diagonal matrix.

I know that a set of vector functions [math]{\vec{v_{1}}(t), \vec{v_{2}}(t)+...+\vec{v_{n}}(t)}[/math] in a vector space [math]\mathbb{V}[/math] if [math]c_{i}= 0 [/math] for the following equation: [math]c_{1}\vec{v_{1}}(t)+c_{2}\vec{v_{2}}(t)+...+c_{n}\vec{v_{n}}(t) \equiv \vec{0}[/math] Where does the Wronskian come into play? Is it basically a determinant with functions and derivatives? Thanks

Are there any nonabelian groups of squarefree order where none of the primes divides any of the others minus 1? If not, is there a proof of this? I've been trying to do this, but the proof for 2 primes doesn't extend, as I can't find how to prove the group with order of the largest prime is normal. =Uncool-

Hey, I'm not the greatest whizz kid at maths but i know my bit. I've recently been having problems with Algebra, Well, To be honest i've never really grasped the true concepts of Algebra, I know a fair bit, Although i get kind of confused on a certain type of equation. Eg. If X=9, Then what is 17X + 7. If anyone can give me a brief explanation on how to solve these and what units to use, That would be awesome and would really help out when my SATS come along in May. Thanks, Ben.