Recent content by JonoPUH

Primitive roots of 1 over a finite field Homework Statement The polynomial x3 − 2 has no roots in F7 and is therefore irreducible in F7[x]. Adjoin a root β to make the field F := F7(β), which will be of degree 3 over F7 and therefore of size 343. The multiplicative group F× is of order 2 ×...

Thank you for your clarification on notation and help!

Homework Statement Let Yn be uniform on {1, 2, . . . , n} (i.e. taking each value with probability 1/n). Draw the distribution function of Yn/n. Show that the sequence Yn/n converges in distribution as n → ∞. What is the limit? Homework Equations So Yn has c.d.f Yn(x) = |x|/n where |x| is...

Thank you very much! You've made my night!

Thank you so much! I think I have it, although it seems very easy, which always seems suspicious to me in maths. Here goes: Ʃ |<x,vj><y,vj>| ≤ √(Ʃ<x,vj>2) √(Ʃ<xy,vj>2) Then by Bessel's Inequality √Ʃ<x,vj>2√Ʃ<xy,vj>2 ≤ √||x||2 √||y||2 So Ʃ |<x,vj><y,vj>| ≤ ||x|| ||y|| as required!

Ok, so according to Wikipedia (I haven't been taught this in lectures), the Cuachy-Schwarz inequality over ℝn is: (Ʃ xiyi)2 ≤ Ʃxi2 Ʃyi2 Do I replace multiplication with inner products? I've tried that, but I must be doing something wrong. (Ʃ <x,vj>)2 ≤ Ʃ<x,x> Ʃ<vj,vj> =...

Homework Statement Let V be a real inner product space, and let v1, v2, ... , vk be a set of orthonormal vectors. Prove Ʃ (from j=1 to k)|<x,vj><y,vj>| ≤ ||x|| ||y|| When is there equality? Homework Equations The Attempt at a Solution I've tried using the two inequalities given to us in...

Thanks for the tip, but could you possibly explain what you mean by the sign of each side? Is it simply that one side is positive, and the other is negative? In which case, I'm not sure how to proceed. Sorry.

I'm still really stuck on this. Is there any more information people require to help me? Thanks.

Homework Statement Suppose that T(x, y) satisfies Laplace’s equation in a bounded region D and that ∂T/∂n+ λT = σ(x, y) on ∂D, where ∂D is the boundary of D, ∂T/∂n is the outward normal deriva- tive of T, σ is a given function, and λ is a constant. Prove that there is at most one solution...

Homework Statement Let V be a finite-dimensional vector over ℝ, and let S and T be linear transformations from V to V Show that n(ST)≤n(S)+n(T) Given Hints Consider the restriction of S to W where W=im(T) Can someone please tell me what the above hint means? I haven't...