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Math Help - Dual Space

  1. #1
    Member kezman's Avatar
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    Dual Space

    Let V = M_n(k) Vector space of n X n Matrices and S \subset V Subspace of Simetric Matrices. Calculate a base for S^o.
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  2. #2
    Super Member Rebesques's Avatar
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    If (E_{ij}) are the basis matrices for M_n, then you know a basis for the dual M_n^*; It consists of the n^2 maps

    \phi_{ij}:M_n\rightarrow \mathbb{R} defined by \phi_{ij}(E_{rs})=\begin{cases}1,\ (i,j)=(r,s)\\ 0, \ (i,j)\neq(r,s)\end{cases}

    Now a symmetric matrix A=(a_{ij}) will satisfy

    A=\sum_{ij}a_{ij}E_{ij}=\sum_{i}a_{ii}E_{ii}+\sum_  {i<j}a_{ij}(E_{ij}+E_{ji}). (1)

    So a basis for S is given by S_{ii}=E_{ii}, \ S_{ij}=E_{ij}+E_{ji}. This means we will need (n^2+n)/2={\rm dim}S mappings for S^*. The decomposition in (1) suggests we define (for 1\leq i\leq n, 1\leq i<j\leq n) \psi_{ii}:=\phi_{ii}, \ \psi_{ij}:=\phi_{ij}+\phi_{ji}.

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

    \psi_{ii}(S_{ii})=1, \ \psi_{ij}(S_{rs})=\begin{cases}1,\ (i,j)=(r,s)\\ 0, \ (i,j)\neq(r,s)\end{cases} so they form the dual basis of (S_{ij}).
    Last edited by Rebesques; August 31st 2007 at 08:02 PM. Reason: sleepy
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