Recent content by Portuga

Proof. To demonstrate that a Bravais lattice (Figure 1) can be considered as a set of points located by a linear combination of primitive vectors of the lattice with integer coefficients, a sequence of claims in increasing order of complexity can be adopted. First, what is shown for one octant...

Thank u very much!

To solve this problem I used two equations: $$ PV=nRT, $$ where ##P## is the pressure, ##V##the volume, ##R##the gas constant, ##T##for temperature and is##n##the number of moles, related to the mass ##m## and molar mass ##M## by $$ n=\frac{m}{M}. $$ It will be also necessary consider the...

I followed the trick of Gram Schmidt ortogonalization method to build an ortogonal basis ##\left\{ u^{\prime},w^{\prime}\right\}## for ##W## and the strategy of decomposing ##v## as ##v_{\parallel}=v\ldotp u^{\prime}u^{\prime}+v\ldotp w^{\prime}w^{\prime}## and ##v_{\perp}=v-v_{\parallel}##. It...

ok, thank you all very much!

I should subtract first component, normalize and then subtract the other one, and normalize it?

I first tried to use a method based on Gram Schmidt orthogonalization method: $$ v_{\parallel}=\left(v\ldotp\frac{u}{\left\Vert u\right\Vert }\right)\frac{u}{\left\Vert u\right\Vert }+\left(v\ldotp\frac{w}{\left\Vert w\right\Vert }\right)\frac{w}{\left\Vert w\right\Vert }, $$ and $$...

Yes, now I got the point.

Oh, thanks!

Well, my guess is that there is something wrong with the factors chosen, because ##\left\Vert \left(0,1,0\right)\right\Vert =1## and \begin{align} \left\Vert F\left(0,1,0\right)\right\Vert &=\left\Vert...

Thank you very much! Sorry for the mess in latex.

Attempt of a solution. By the Rank–nullity theorem, $$ \dim V=\dim Im_{F}+\dim\ker\left(F\right) \Rightarrow n=1+\dim\ker\left(F\right) \Rightarrow \dim\ker\left(F\right)=n-1. $$ Similarly, it follows that $$\dim\ker\left(G\right)=n-1.$$ This first part, for obvious reasons, is very clear. The...

Oh my god! It was so simple! Thank you very much!

This is actually a solved exercise from a Brazilian book on Linear Algebra. The author presented the following solution: The kernel and image theorem tells us that dimension ##\dim V=n=\dim\ker\left(F\right)+\dim \text{im}\left(F\right)=\dim\ker\left(G\right)+\dim\text{im}\left(G\right)##. As...

Ok, thank you very much.