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