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...
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...
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
$$...
Well, my guess is that there is something wrong with the factors chosen, because ##\left\Vert \left(0,1,0\right)\right\Vert =1## and
\begin{align}
\left\Vert F\left(0,1,0\right)\right\Vert &=\left\Vert...
Attempt of a solution.
By the Rank–nullity theorem,
$$
\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...
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...