To find dual basis from the inner product Matrix?

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SUMMARY

The discussion centers on constructing a dual basis from an inner product matrix, denoted as Γ (Gamma). The user acknowledges that an orthonormal basis serves as a dual basis and mentions the Gram-Schmidt method for constructing orthonormal bases. However, they express uncertainty about deriving a dual basis directly from the inner product matrix Γ. The conversation indicates a need for clarity on the relationship between inner product matrices and dual bases.

PREREQUISITES
  • Understanding of inner product spaces
  • Familiarity with dual bases in linear algebra
  • Knowledge of the Gram-Schmidt orthonormalization process
  • Basic concepts of linear transformations
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  • Learn about the relationship between dual bases and inner product spaces
  • Explore advanced applications of the Gram-Schmidt method
  • Investigate the role of linear transformations in constructing dual bases
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Students and educators in linear algebra, mathematicians interested in dual spaces, and anyone seeking to deepen their understanding of inner product matrices and their applications in vector spaces.

Nile@ua
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To find dual basis from the inner product Matrix!?

Homework Statement


WE know the inner product matrix (capital)Gamma and that's all. How do we "construct" a dual basis?



Homework Equations





The Attempt at a Solution



I know that the orthonormal basis is nothing but a dual basis. and we can constuct orthonormal basis using G-S method. But, I am clueless about how to construct dual basis from Gamma?

Thanks in advance.
 
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I got it, if anybody wants i can post the answer. :-)
 

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