Understanding the Reciprocal Basis Problem

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Homework Help Overview

The problem involves understanding the concept of reciprocal bases in a vector space, specifically relating to the standard basis {i, j, k}. The original poster seeks to demonstrate that the reciprocal basis {i*, j*, k*} is equivalent to {i, j, k} using the definitions provided.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • The original poster expresses uncertainty about how to begin the problem and questions the meaning of {i*, j*, k*}. Other participants inquire about the notation used, specifically the meaning of [a, b, c] and whether a x b refers to a cross product or tensor product.

Discussion Status

The discussion is ongoing, with participants seeking clarification on notation and definitions. There is no explicit consensus yet, as the original poster's question remains largely unanswered, and multiple interpretations of the notation are being explored.

Contextual Notes

Participants note that {i, j, k} is a set of vectors rather than a matrix, which may influence the understanding of the problem. The notation and definitions used in the problem statement are under scrutiny, indicating potential gaps in the original poster's understanding.

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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}?
 
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Did anyone not understand my question?
 
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.
 
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]?
 
HallsofIvy said:
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.
 

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