Recent content by porroadventum

Homework Statement Show that the sequence (xn)n\inN \inZ given by xn = Ʃ from k=0 to n (7n) for all n \in N is a cauchy sequence for the 7 adic metric. Homework Equations In a metric space (X,dx) a sequence (xn)n\inN in X is a cauchy sequence if for all ε> 0 there exists some M\inN such...

Homework Statement Prove Z[x]/pZ[x] is an integral domain where p is a prime natural number. Homework Equations I've seen in notes that this quotient ring can be isomorphic to (z/p)[x] and this is an integral domain but I don't know how to prove there is an isomorphism between them and...

THat's great, thank you both for your help

Homework Statement Let A be the following 3 × 3 matrix: A =([4 2 6],[2 1 3],[2 1 3]) i) Find the rank of A ii) Show that there exists an 3 × 2 matrix W, of rank 2, such that AW = 0. iii) Construct one such matrix W. Homework Equations The Attempt at a Solution I think the answer to part 1 is...

OK, I see how to prove it now- the addition of two elements of S will always give a number which is a multiple of a, therefore will be an element of S. (Similarly for multiplication)...? (Thank you)

Well all real numbers are lie within the complex set of numbers so you could define a function as σ:ℝ→ℂ such that σ(a)= a+0i where a \inℝ. This is a bijective map. Now you just need to show that this function preserves the group operation...

Homework Statement Consider the ring Z/mZ, show that S = {[0], [a], [2a], · · · , [m − a]} forms a (possibly nonunitary) subring of Z/mZ when a divides m. (i.e. show that (S,+, ·) is closed the usual addition and multiplication. (We are not require to find a multiplicative identity)...

I have made an attempt, if someone could let me know if it is correct or not, that would be much appreciated! The elements of Z/9Z are {0,1,2,3,4,5,6,7,8} with operation modulo9. The elements (1,2,4,5,7,8} have order 9 and generate the whole group. {0,3,6} has order 3. Therefore there...

Homework Statement Describe all the subgroups of Z/9Z. How many are there? Describe all the subgroups of Z/3ZxZ/3Z. How many are there? The Attempt at a Solution I don't even know where to start with this question. If someone could just point me in the right direction that would be...

I'm getting the minimum at a value of 2√3, is this correct?

Homework Statement Find the minimum of f(x,y)= 3x2+y2, subject to the constraints 1<=xy. Homework Equations I thought I would use Karush Kuhn Tucker's theorem to solve this. ∇f=(6x, 2y) and ∇h=(-y,-x) The general equation according to KKT is ∇f=λ∇h. First case: h<0. According to...

Homework Statement Let Ʃ from n=1 to ∞ an and Ʃ from n=1 to ∞ bn be convergent series, with an\geq0 and bn\geq0 for all n\inN. Show that the series Ʃ from n=1 to∞ max(an,bn) converges. Homework Equations I'm guessing it's got something to do with the cauchy criterrion for convergence...

Does that mean the interval of convergence is (-1/4,1/4)? Because when I let x=1/4,=-1/4 I get 1 and +/-1 respectively which are divergent series. Therefore it is absolutely convergent in (-1/4,1/4) and is divergent everywhere else. THank you for your help on this.

Oh so it will the orginal series will have a radius of convergence of 1/4?!

...3/4?