Recent content by lukaszh

Hello, why the set of all real numbers is complete metric space with euclidean metric? I know, that metric space is complete iff all sequences in it converges. But 1,2,3,4,... diverges. Thanx

If such finite cover exists, then for some N \mathcal{A}\subset\bigcup_{n=1}^{N}\left(\frac{1}{n},1\right)=\left(\frac{1}{N},1\right) That's not true, because there exists \frac{1}{N+1} which is not covered. ThanX :-)

aha, thanx. So, for example for cover \bigcup_{n=1}^{\infty}\left(\frac{1}{2^n},\frac{1}{2^n}+\frac{1}{10}\right) there's no finite subcover. Is it correct?

This is not true in general: \sqrt{x}\sqrt{y}=\sqrt{xy} !

I can't understand why the set \mathcal{A}=\left\{\frac{1}{2^n};\,n\in\mathbb{N}\right\} is not compact, while \mathcal{A}\cup\{0\} is. I know that set is compact if and only if it's closed and bounded, so in order to make set \mathcal{A} closed, we need to include zero, as it's condesation...

12=1 (mod7) 22=4 (mod7) 32=2 (mod7) 42=2 (mod7) 52=4 (mod7) 62=1 (mod7) 72=0 (mod7) Is it periodic {1,4,2,2,4,1,0} ? Now I know :-) It's periodic, so if I add any of these congruences together there will be some remainder. Remainder is zero if and only if I add congruences in form (7k)2=0 (mod7)...

Hello, how to prove If 7|(a^2+b^2) then 7|a and 7|b. (If seven divides a^2+b^2 then seven divides a and seven divides b) Thanks.

hi, what is wrong about this proof? If x^TAx=0 then A is antisymetric matrix. True? false? P: False A=-A^T x^TAx=-x^TA^Tx x^TAx=-(Ax)^Tx x^TAx=-\lambda\Vert x\Vert^2 If x^T.A.x is zero, then must be -\lambda\Vert x\Vert^2, but ||x|| is real nonzero number and lambda must be zero. But...

Beautiful, thank you.

[FONT="Century Gothic"]Hello, so, I know that A=S\Lambda S^{-1} Eigenvalues of A are \{\lambda_1,\lambda_2,\cdots,\lambda_n\} for n\times n matrix. If I add to both sides identity, then A+I=S\Lambda S^{-1}+I=S\Lambda S^{-1}+SS^{-1}=S(\Lambda+I)S^{-1} Its determinant is...

I'm so sorry, because I've made a mistake. Prove that det(A+I)=1. Thank you again for your reactions.

Thank you. Now I understand. Thanks Thanks

Hello, I don't want how to prove: Matrix A is nilpotent, so A^k=0. Prove that det(A+I)=0. Thank you so much :-)

Hello, everywhere I can see this a_n = \frac{1}{\pi}\int_{-\pi}^\pi f(t) \cos(nt)\, dt b_n = \frac{1}{\pi}\int_{-\pi}^\pi f(t) \sin(nt)\, dt etc... I can't find, how to derive this formulas. I'm really tired and a bit confused of this formulas, because I can't find possible way to derive...

Hello, i can't find anywhere,what is and how to find matrix signature. wikipedia tells only, that signature matrix is matrix with +/-1 on diagonal. For example 1 1 1 1 1 1 1 1 0 how to find signature. THank you :-)