Calculating reciprocal base vectors

[epr1990]

I have just started diving into tensor analysis. To be honest, I didn't know whether to post this question in the vector analysis forum or this one. I have looked at a few books on the subject and scoured the internet, but I can't seem to find anything that answers this question. Or, maybe I just didn't understand what was being said.

Say you are given a set of basis column vectors in n dimensions [itex]\{\vec{g}_{i}\}[/itex], for i=1,...,n and let [itex]G=[\vec{g}_{i}][/itex] be the matrix of column vectors. Can the reciprocal base vectors be calculated simply by taking the inverse of this matrix where [itex]G^{-1}=[\vec{g}^{i}][/itex] is the matrix of the row vectors representing the reciprocal basis [itex]\{\vec{g}^{i}\}[/itex]? This would make since because matrix multiplication would yield the identity matrix, and, when written in this form, matrix multiplication becomes the inner product of the vectors that construct the respective matrices as so:

[itex]G^{-1}G=[\vec{g}^{i}][\vec{g}_{j}]=[\vec{g}^{i}\cdot\vec{g}_{j}]=[\delta^{i}_{j}]=I[/itex]

Since the inner product of two roof or two cellar base vectors isn't necessarily the kronecker delta, this is the only way that makes sense to me. So, in other words, the rows of the inverse of G would be the reciprocal basis. Is this correct?

Also, I have gotten through most of Simmonds book A Brief Introduction to Tensor Analysis. For much of the book, he uses the less formal terms roof and cellar instead of Covariant and Contravariant. Are these latter terms only applied to the components of a vector? Or can they be applied to the vectors themselves?

 

[bolbteppa]

Try & get your hands on the last two chapters of Eutiquio C. Young's book 'Vector and Tensor Analysis', you'll see him work some baby examples of calculating reciprocal bases first from the definitions, then using cross product simplifications when in R^3, then again using matrices, & developing the basic theory behind these approaches, so you get multiple angles & nice examples to codify what you're doing, & I can't find anything better than this. You're right about the reciprocal basis being given by the inverse matrix, & the book will give a nice simple argument why this is the case similar to yours. Also you can refer to either the vector or the components by those terms & not commit any egregious sin, I think I've come across some sources arguing that referring to one or the other by these terms makes no sense, but so many books refer to both by these terms that it's basically an accepted abuse of notation if one is logically incoherent.

 

[epr1990]

Thanks.

I actually got through and understood everything in Simmonds without knowing that. It was just a problem I came across when going back and doing the exercises. I guess it wasn't explicitly stated, but it was supposed to be inferred at some point. I failed to realize that I actually had been doing that all along just in component form.