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.

2 Comments

Filed under algebra, calculus on manifolds, cucei math, differential geometry, math analysis, multilinear algebra, tex

2 responses to “duality rules!

  1. I agree a couple of points I would like to add:
    1) It might be pointed out that the notation is just a syntactically consistent notation to discuss mapping from domain to domain (like you did point out). i.e. a language that allows discussions.
    2) This can be expanded and illustrated by the metric tensor from differential geometry; i.e. discussing the same physical objects in both polar and rectangular coordinates.
    3) “wedge product”; I never understood all the flim-flam about matrix minors and such until I understood wedge products. I think this simple algebra grounding would motivate learning wedge products. It’s true that there is an axiomatic system; but it seems quite abstract when first introduced whereas showing it as an powerful (and abstract) form of matrix minor properties illuminates a lot in both directions.
    Of course the “wedge product” abstraction has a lot more power in talking about dependent subspaces and such.
    4) In the simple abstraction co-vectors are just another form of vectors; until you start talking about coordinate transforms and exterior derivations. They do have there own dual space which may or may not be the original vector space. A case where the dual or the dual is not the original is the modern Umbral Calculus; but the very fact that these ideas can be used in function spaces as well as “ordinary” vector spaces shows the strength of learning and using these ideas.

    Ray

    • Thanks Ray, of course I’m over-emphasizing the nonexistent ugliness of vector calculus, I only use this (insulting) technique to highlight the urgency to change how the ideas of calculus and linear algebra are addressed by the science schools. A small step towards a higher academic status of a young mind is transiting the Grassmann algebra of finite dimensional spaces over the real numbers. And I am glad you shared how these ideas paved the way to higher dimensions and then upgraded us.

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