# duality rules!

in this post I am gonna again insist about the cruel confusion spawn by people who, I think, misunderstand certain strategic points in the grasping of tensors.

Vectors in $\mathbb{R}^n$ are linear combinations of

$e_1=\left(\begin{array}{c}1\\ 0\\ 0\\\vdots\\ 0\end{array}\right)$ , $e_2=\left(\begin{array}{c}0\\ 1\\ 0\\\vdots\\ 0\end{array}\right)$ , … , $e_n=\left(\begin{array}{c}0\\\vdots\\ 0\\ 0\\ 1\end{array}\right)$ ,

that is, vectors are expressions as $v=v^1e_1+v^2e_2+\cdots+v^ne_n$, or employing the Einstein-Penrose convention: $v=v^se_s$.

But, covectors are linear combinations of

$e^1=[1,0,0,...,0]$ , $e^2=[0,1,0,...,0]$ , … , $e^n=[0,...,0,1]$ ,

which are also dubbed “projectors” due its role as a linear maps ${\mathbb{R}}^n\to{\mathbb{R}}$.

So a general covector is a linear combination $f_se^s$, using the Einstein-Penrose Convention again.

It is regrettable how many authors are merciless in the misuses of the rows and columns concepts for vectors and covectors respectively. It seems that sacrificing the difference by writing vectors as rows to gain space in the texts of linear algebra books is a big cost in the habits of understand well the matter.

This is perhaps a simple reason why it is so difficult to everybody to grasp the idea of tensors at its first steps. Well, not everybody … hehehe.

We say that duality rules ‘cuz

$e^i(e_j)={\delta^i}_j$

becomes a simple device to serve as a guide for a (apparently) complex calculations that arise when one consider tensors of higher rank.

Once recognized this hurdle, the constructions of a calculus à la  Cartan, using the wedge product and the exterior derivative among tensor fields is an easy cake. These techniques will allow you to free of the ugly vector calculus that rules in each bachelor science schools nowadays.