Show that the reciprocal basis of the standard basis is itself

[Benzoate]

Homework Statement

Let {a,b,c} be any basis set. Then the corresponding reciprocal {a*,b*,c*} is defined by
a*=b x c/[a,b,c] , b*=c x a/[a,b,c], c*=a x b/[a,b,c]

If {i,j,k} is standard basis, show that {i*,j*,k*}={i,j,k}

Homework Equations

The Attempt at a Solution

I have no idea how to start this problem. I know the standard basis is just the identity matrix. But that's all I know. I don't know what {i*,j*,k*} is supposed to symbolized. Is it the inverse of {i,j,k}?

 

[Benzoate]

Did anyone not understand my question?

 

[Ben Niehoff]

What is the notation [a,b,c]? A vector triple product?

Also, {i,j,k} is not a matrix. It is a set of three vectors.

 

[HallsofIvy]

First you are going to have to clarify your notation. Is a x b the cross product of two vectors or is it a tensor product? And what do you mean by [a, b, c]?

 

[Benzoate]

First you are going to have to clarify your notation. Is a x b the cross product of two vectors or is it a tensor product? And what do you mean by [a, b, c]?

a x b is the cross product [a,b,c] is the basis set.