Linear Algebra and Group Theory

I am surprised at how small amount of information I could find on this subject when surfing the web. My teacher a while back showed us a clever little quick way to find the sqaure root of a number, does anyone know how to do this?{if so, please list} Thanks.

Given that the definition of a function f, from A to B (sets) is: 1) f is a subset of AxB............................................................................... [math]f\subset AxB[/math] ,and 2)for all xεΧ ,there exists a unique yεΥ SUCH that (x,y)εf .......................................... [math]\forall x [x\in X\Longrightarrow\exists! y(y\in Y\wedge (x,y)\in f)][/math] prove that the following is a function from A to B,where A = {1,2} B = { a,b } f = { (1,a),(2,b) }

1. prove that the value of a determinant is unique? ie the ans is same whichever way we expand it starting from whichever column? 2. det(AB)=det(A)det(B) why? where A,B r determinants of order n? 3. for a matrix to be orthogonal (A transpose)(A)=(A)(A inverse). again if AB=AC we cant say that B=C. so how can we say in the previous case that A transpose=A inverse? 4. is identity matrix unique can be proved? ie if AB=A can we say B is unique? if yes how? thanks in advance.

Trying to prove that the proper orthogonal group SO(3) acts transitively on the set of points on the surface of a sphere. Can show that if you assume 2 points on the sphere can be related by x=Ay where A is a 3x3 matrix then A multiplied by its transpose must be I but this only shows A would be in O(3) not SO(3) - i.e. how can you show any 2 points can be linked by A where det(A) = 1? Any help would be much appreciated!

What theorem or axiom allow us to do the substitutions : x= a + b ,or y= a + b + c, or x= 2y + z^2 e.t.c , e.t.c some times in a mathematical proof ??

This is the first question: http://img195.imageshack.us/img195/2949/64542292.jpg This one include three questions http://img268.imageshack.us/img268/1953/345x.jpg Let's show work out =) Good Luck.

I am a bit confused by this. Here is the equation: [math] a=b[/math] [math]a^{2}=b^{2}[/math] [math]a^{2}-b^{2}=0[/math] [math](a+b)(a-b)=0[/math] [math]a=b,-b[/math] let [math]a=1[/math] [math]1=1,-1 [/math] I understand that this cannot be true but why does it work algebraically? To my understanding, if [math] a=b [/math], than [math] a^{2}=b^{2}[/math] but if [math] a^{2}=b^{2} [/math], than [math] a\neq b [/math] What am I not understanding?

given a set with the symbols : " + " for addition " - " for inverse the constants : 1 0 AND the axioms: for all a,b,c : a+(b+c) = (a+b) + c for all a : a+0 = a for all a : a +(-a) =0 for all a,b : a+b = b+a Give a rigorous proof of the following: 1) 0 is unique 2) the inverse of a ,-a is unique 3) for all x,y : -( x+y) = -x +(-y)

I need the help of a math's wizard, because I am confused by something.... The odds of winning £10 on the Lotto are 56.7 to 1. Ok. But I have been told that if you write 57 lines of numbers, no matter what they are, you can't always win £10. In fact I have been told that you need 168 lines of numbers to guaranty £10. Could anyone explain the mathematics behind this in simple terms please. I am trying to write a computer program to find the smallest number of lines possible. I also want someone to be able to type 6 numbers into the computer, and the computer will make the smallest number of lines using those numbers. Pincho

here is a question that we were asked in electronics class a few years ago, we`de just learned Ohms law. Non of us ever managed to work it out here goes..... Imagine a cube made of (100 Ohm) resistors. so that would be a total of 12 identical resistors. work out using ohms law, the resistance between the front left bottom corner, and the top right back corner. A diagonal thing. each edge is a 100 Ohm resistor. I don`t think it`s possible myself using just ohms law, but would love to be shown either way, and certainly the answer!

I've been working on a problem that turns out to have a group structure. Group is of order 512. Abelian, non cyclic. Would like to find which group. I see that there are 10 or so Abelian groups of order 512. Any where I can can find a catalogue of these so I can find which one I have. Or some process to work through to that will classify the group?

Hi, Is there any solution for the following problem: [math]Ax = \lambda x + b[/math] Here [math]x[/math] seems to be an eigenvector of [math]A[/math] but with an extra translation vector [math]b[/math]. I cannot say whether [math]b[/math] is parallel to [math]x \quad[/math], ([math]b = cx[/math]). Thank you in advance for your help... Birkan

Dear All, I'm going through a project with my university classmates but we are asked to evaluate each other's contribution by assessing our effort in a peer evaluation form. In particular, we are asked to: rate our collegues' contributions by assigning: . maximum 100 points per person . each one has to assign exactly 300 points . not to assign any points to ourself . differences between any score we assign to our mates has to be equal or superior to 5 . it can't be the case in which everyone has the same total score (but may be slightly different!) we will receive a final note based on the note we receive as a group and, individually, by correcting it for…

Hello, hope some one can help me on this. I am writing a piece of code that generates a square matrix A: Do j=1,n Do i=1,n A(i,j)=A(i,j)+const ... end do end do Now I have to calculate A^2. If I don't want to call a subroutine to do that but want to finish this at the same time when I am generating the matrix A, is there a way to do this? Thank you very much.

If a group has order 12, must it have an element of order 2? Why? I am relatively new to group theory and I have to admit this question stumped me. Does anyone have an idea?

my brother has some math problems where he has to solve by graphing things like x+2y=10 3x+4y=8. i know this is basic easy algebra, but i covered this stuff long ago and am more of a biology person anyways so i dont deal with much math at all anymore. can someone just give me the basics on how to do this? thanks.

Dear all, I'm a math student and i facing problem to solve these question.hope all of u can help. 1. what would be wrong with defining matrix multiplication for matrices of the same size by multiplying them entry-by-entry, as with addition and subtraction? 2. discuss the relative merits of computing determinants between cofactor expansion method and row reduction method. 3. under what conditions will a matrix be diagonalizable? Are there any cases where one can tell at a glance whether a matrix is diagonalizable? hope someone can help me...thanks a lot..

What are matrices actually used for practically? (if at all)

I have a quick question regarding the identification of the lowest upper bound and the greatest lower bound for the zeros of a polynomial function. The method that my book shows is to use to upper and lower bound theorem involving synthetic division. This involves performing division with 1, 2, 3, etc. and -1, -2, -3 etc. until the bounds are found, but what if the lowest upper bound is something like 57 or 89? I'm I supposed to perform division for every number or is there a way to identify a smaller group of possible bounds. I've just been using the results of division to estimate about where it would be and eliminated possibilities from there, it just does not see…

I seem to be having an inordinate amount of difficult with understanding the concept of horizontal stretching and compressing graphs. Vertical stretching and compressing makes sense. [math]y=2f(x^2)[/math] If I understand it, this is simply multiplying the output of the function by two. All that happens is that the [math]y[/math] value is double whatever it would be as a result of [math]y=f(x^2)[/math]. This means that an input of 1 gives a [math]y[/math] value of 2, an input of 2 gives 8. This would stretch the graph away from the x-axis due to the y value increasing faster than it otherwise would. The part that confuses me is when you multiply the [math]x[/math…

Hi, I'm trying to find a simple algorithm/sourcecode for the generalized eigenvalue problem, which means finding eigenvectors for: Ax = \lambda Bx I'm aware to the QZ algorithm, but since I'm only intrested in the 3x3 matrices case, I wonder if there's an easier way. thanks in advace, Vince

Given a linear transform T:V->V It has to be proven that if ker(T^2) = ket(T)than Im(T^2) = Im(T) and (the opposite) if Im(T^2) = Im(T) than ker(T^2) = ket(T) Thank you in advance for the help

Hi, I am designing an animation library for the web, and I wanted to create an animation interpolation that is more flexible and elegant than using 'sine' or similar. I have spent countless hours trying to solve this, so I'll be extremely thankful for any help offered. My system I devised is this: A partical is at position 0, with velocity 'v' and accelleration 'a'. I want it to move along a function to land at position 'h' with velocity 0 and accelleration 0. Simply put, I want to get a moving object to land somewhere stationry. 'v' and 'a' are provided to the function as attributes so that if the target moves then the curve can adjust without any abr…

Hi everybody, I've been doing linear algebra revision lately and one query I have has really been winding me up so I'd be really grateful for some help: If I have a linear operator that is not Hermitian or unitary, will its eigenvectors necessarily be orthogonal(in absence of degeneracy)? In either case why? Also, if the eigenvectors do have to be orthogon when there's no degeneracy would you be able to show how this is true? Thanks a lot in advance:-)