To find dual basis from the inner product Matrix?

In summary, the dual basis can be found from the inner product matrix using the Gram-Schmidt process, which involves orthogonalizing the columns of the matrix to create an orthonormal basis. This is important because it allows for the representation of linear functionals in terms of the inner product matrix, which has various applications. The concept of a dual basis can be applied to any vector space with an inner product, and the inner product matrix is used to find the dual basis. Alternative methods such as Cholesky or QR decomposition can also be used to find the dual basis.
  • #1
Nile@ua
2
0
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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  • #2


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

1. How do I find the dual basis from the inner product matrix?

To find the dual basis from the inner product matrix, you can use the Gram-Schmidt process. This involves orthogonalizing the columns of the inner product matrix to create an orthonormal basis. The dual basis can then be found by taking the transpose of the orthonormal basis.

2. What is the importance of finding the dual basis from the inner product matrix?

The dual basis is important because it allows for the representation of linear functionals in terms of the inner product matrix. This can be useful in various applications, such as optimization problems and solving systems of linear equations.

3. Can the dual basis be used in any vector space?

Yes, the concept of a dual basis can be applied to any vector space that has a defined inner product. This includes finite-dimensional vector spaces as well as infinite-dimensional spaces such as function spaces.

4. How does the inner product matrix relate to the dual basis?

The inner product matrix provides a way to calculate the inner product of two vectors in a vector space. This inner product can then be used to find the dual basis, which is a set of linear functionals that are orthonormal with respect to the inner product.

5. Are there any alternative methods for finding the dual basis from the inner product matrix?

Yes, there are alternative methods such as using the Cholesky decomposition or the QR decomposition to find the dual basis. These methods may be more efficient or numerically stable in certain situations.

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