Linear Algebra and Group Theory

When defining the dot product, the cosine of the angle between two vectors is chosen. Why not the sine? What advantages are there by choosing the cosine?

for the following problem: For all ,a : if [math]0\leq a\leq 1[/math], then [math]\sqrt{1+\sqrt{1-a^2}} +\sqrt{1-\sqrt{1-a^2}} =\sqrt{2+2a}[/math].....................................................................................A the following proof is suggested: [math]0\leq a\leq 1\Longrightarrow 0\leq 1-a^2\leq 1\Longleftrightarrow 0\leq \sqrt{1-a^2}\leq 1[/math][math]\Longrightarrow(\sqrt{1+\sqrt{1-a^2}}\geq 0\wedge \sqrt{1-\sqrt{1-a^2}}\geq 0)\Longrightarrow[/math] [math](\sqrt{1+\sqrt{1-a^2}}+\sqrt{1-\sqrt{1-a^2}})^2[/math] = [math]1+\sqrt{1-a^2}+1-\sqrt{1-a^2}+2\sqrt{(1+\sqrt{1-a^2})(1-\sqrt{1-a^2})}=2+2|a|=2+2a[/math] And since [math] 2+2a\geq 0[…

Hi, Does one of the conditions for the definition of Vector Space V over a Field F include : 1) Closure property in vector addition. 2) Closure property in scalar multiplication. Or are these conditions included only in the definition of Linear Vector Spaces? Clear Skies! kSqUARe

If v1 = u1 + u2 + u3, v2 = u1 + a*u2, and v3 = u2 + b*u3...where u1, u2, and u3 are given linearly independent vectors, find the condition that must be satisfied by a and b in order to ensure that v1, v2, and v3 are linearly independent.

Hello, this is my first thread and first post. I was bored today and came up with this interesting problem that I cannot solve (probably to do with the fact that I'm only in calculus II). We are going through sequences so this is my inspiration. So here it goes. (Note: I'll be using a sub n in place of n being a subscript of a because I don't know how to do that on this forum.) Let a be a sequence with a sub n being the nth term. Let a sub n=n^(n^x)/(n!)^n. Find max. a sub n in terms of x where x satisfies that the limit of a sub n as n approaches infinity equals zero and n is in the natural number set. Have fun.

I advance two similar S-hypotheses (“Sorokin’s hypothesis”). S-hypothesis-1 (confirmed by concrete calculations on the computer): In the equality [math]A^{n^t}+B^{n^t}=(A^{n^{t-1}}+B^{n^{t-1}})R[/math], where - prime n> 2, - integers A and B are relatively prime, (t-1)-digit end of each prime divider of the number R (excluding n) in the base n is equal to 1. For example (for n=3): [math]8^3-1=(8-1)*73=(8-1)*(8*9+1) [/math]; [math]8^3+1=(8+1)*57=(8+1)*(3*19)=(8+1)*[3*(2*9+1][/math]; and so forth. S-hypothesis-2 (more questionable): In the equality [math]A^n+B^n=(A+B)R[/math], where - prime n> 2, - integers A and B are relatively prime, - t…

My math teachers say that 0 to the power of 0 is undefined, but after messing around a little, I found this: 00=00 Original equation 00=00*1 Multiply one side by one 00/00=1 Divide both sides by 00 00=1 Simplify using the division laws of exponents Are my teachers incorrect or did I make a mistake?

Cant seem to fathom what an R-module is. Thing is i can prove if something (elementary structures for now) is an R-module or not, but i dont get what the basic structure 'looks' like. Let me explain... Graphs and rings are fine, in being sets which satisfy certain criteria. So i understand that they are special sets, operative word SETS, a structure i understand. R-modules on the other hand utilise these special sets (abelian groups, rings), uses an endomorphism onto, say, the group 'space', then end up being a structure i vaguely understand (obviously a set itself, but different somehow to me). So for now i view an R-module as a ring (again, satisfying certain axioms…

Ola everyone, kinda need help with Hp 35s scientific calculator...when I am about to calculate the sample standard deviation from values that are weighted I just can not get the right result wotever i do...it really drives me crazy...it calculates samfin but when I compare the output with the output from SPSS /statistics program/, it is just incorrect, and I follow the device manual book. pls if anybody helped, i would be most grateful.

Hey everyone, I'm taking this course called "Number Theory" and am having a lot of difficulty with it. We're currently on "proofs" and i am having some issues. Last week was my first weeks classes. The first day my professor jumped right into the material without giving much background. He's assigning problems left and right without giving any class examples, and the textbook seems like more of a novel than a math book. I'm stuck on one problem. I "think" it's asking for a proof, but the directions are very unclear. It asks: "Tell whether each statement is true and give counterexamples to those that are false. Assume a, b, and c are arbitrary nonzero i…

I have constructed augmented matrix [math] \left[ \begin{array}{ccccccc|c} \frac{4}{5} & -1 & 0 & 0 & 0 & 0 & 0 & 1\\ 0 & \frac{4}{5} & -1 & 0 & 0 & 0 & 0 & 1\\ 0 & 0 & \frac{4}{5} & -1 & 0 & 0 & 0 & 1\\ 0 & 0 & 0 & \frac{4}{5} & -1 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 & \frac{4}{5} & -1 & 0 & 1\\ 0 & 0 & 0 & 0 & 0 & \frac{1}{5} & -1 & 0\\ \end{array}\right] [/math] and I would like to find all solutions. How would I go about doing that?

Hey guys, I'm in the class "Number Theory" at my college. It's WEEK 1 and we are already going over these weird proofs and i am COMPLETELY lost. The question asks: "Show that if n is any odd positive integer, and m = (n^2-1)/2, then m^2 + n^2 = (m+1)^2." My professor as awful and has given us absolutely NO background. He's given us two proof examples in class, but i don't understand either of them, so they don't help much. From what i can gather, odd numbers are "2k+1" and even numbers are "2k". I guess this makes sense, but i don't understand how to use it. So for the problem on hand, i understand i need to show how you get from point A to point …

Started off with graph theory with some elementary things. There's one thing however that i cant wrap my head around, the order of the edges. I get that if the vertices of a graph were included in a set S, the number of edges is equal to the order of the power set of S. For the life of me i cant remember the proof if i did it or not. Can someone please provide the proof of |P(S)|=2^(nC2), where P(S) is the power set of set S, |S|=n and C refers to the combination function? Much appreciated

I've just been reading The Emperor's New Mind by Rger Penrose (A damn fine book, and one you should read) and it mentions a theorem put forward by a chap called Godel. Basically, what he showed was, any precise system of mathematical axioms and rules of procedure, provided that it is broad enough to contain descriptions of somple arithmeic propositions, and provided that it is free from contradiction must contain some statements that are neither provable nor disprovable by the means allowed within the system. This got me thinking about the universe generally, which can be represented as a mathematical set of rules... would the same apply? would there be things that can…

Given a matrix in the following form [math] H \begin{bmatrix} 0&1 & \\ 1&0& 1 \\ &1 &0 & 1 \\ &&\ddots &0& \ddots\\ &&&1&0 & 1 \\ &&&&1&0 \\ \end{bmatrix}_N [/math] ,how to prove that the eigen value of the matrix H are ? [math] E=2\cos\left( \frac{n\pi}{N+1} \right); \quad n = 1, 2, \dots , N [/math] Help:confused::confused:

I cannot get round the attached practice set of questions. Does anyone have solutions please? problem.doc

I'm struggling with the attached practice questions. Could any one with category theory knowledge please help with a solution question.doc

The series: [math] X_n = e^{iX_{n-1}}[/math] seems to converge to [math] .576412723 + .3746990207i [/math] no matter what [math] X_0[/math] is. Can anyone explain why this is?

Hey everyone, I was just wondering if a piece of graphite that has 60 carbon atoms can form a closed polyherdral structure? I think that the key to answering the this is using eulers forumla but I dont know how to use it. this is as much I can do; V + F = E + 2 60 C atoms divided by 5(for vertices of pentagon) will give 12 pentagons. 12 pentagons make 60 vertices. 5 hexagons can be linked to a petagon to give a closed structre and each heaxagon is then linked to 3 pentagons and three hexagons. I cant go further than this. Any help much appreciated. thanks.

Hey all! I have tried to solve this problem and know part of it is right, i'd appreciate it very much if you could take a look at it. Determine the kernel and range of the transformation T and find a basis for each: T(x,y,z)=(x+y, z) from R3 to R2 I have so far that the kernel is the set {(r, -r, 0)}. The range is R2. What is the basis? And is the range right? Thanks for your help!!!

In a book I have the following step isn't clear to me:- r = m x sqrt(kT/m) ------> r = sqrt(mkT) I just can't see the steps that go from the first equation to the second. Could anyone show me the missing step(s)?

To prove : for all natural Nos ,n : [math]n^2 [/math] is even[math]\Longleftrightarrow[/math] n is even.....................A The following proof was suggested: for all ,nεN : n=2k or n= 2k + 1 ......κεN But ,[math] n^2 = (2k + 1)^2 =4k^2 + 4k +1 = 2(2k^2 +2k) + 1[/math] Hence : [math]n^2[/math] is not even or n is even....................1 Also: [math] n^2 =(2k)^2 = 2(2k^2)[/math] Hence: [math]n^2[/math] is even or n is not even......................2 From (1) and (2) we conclude: (A)

My textbook isn't really clear on the method of finding an equivalent matrix. Is there a fixed set of elementary operations to be performed, or is it done purely by trial/error and logic? Thanks.

can somebody give the definition of rigorous proof?? How for example one would write a rigorous proof for: (-x)(-y) = xy for all x,y. thank you

The graph of a quadratic function f(x) has its vertex at (5,-6) and passes through the point (2,-51). Find a formula for f(x) I got 15(x-5)^2 - 6 Why is this wrong? Find the minimum value of the function (5x-2)(2x+2). I got -49/20 after multiplying out the equation to get 5x^2+3x-2 (simplified). I did -3/10 and put it back into the equation and got it wrong. . . Any ideas?