Let Vector space of n X n Matrices and Subspace of Simetric Matrices. Calculate a base for .

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- August 31st 2007, 11:15 AMkezmanDual Space
Let Vector space of n X n Matrices and Subspace of Simetric Matrices. Calculate a base for .

- August 31st 2007, 08:57 PMRebesques
If are the basis matrices for , then you know a basis for the dual ; It consists of the maps

defined by

Now a symmetric matrix will satisfy

. (1)

So a basis for is given by . This means we will need mappings for . The decomposition in (1) suggests we define (for ) .

Now you can prove these are linearly independent and thus form a basis. In fact, they satisfy

so they form the dual basis of .