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Math Help - identity linear operator

  1. #1
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    identity linear operator

    Let I: V->V be the identity operator on an n-dimensional vector space V defined by I(v)=v for every v in V. Show that the matrix of I with respect to a basis S for V is I_{n}.
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  2. #2
    Super Member Gamma's Avatar
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    Well it is the identity. So in particular, it takes the first basis element to the first basis element ( 1e_1 + 0e_2 + ... + 0e_n), the second basis element to the second basis element ( 0e_1 + 1e_2 + ... + 0e_n), etc.
    Thus the first column is [1,0,0...0]
    The second Column is [0,1,0,...0]
    .
    .
    .
    The nth column is [0,0,...,1]
    So a_{ii}=1 and a_{ij}=0 for  i \not = j
    a_ij=\delta_{ij}=In
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