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Math Help - inner product spaces

  1. #1
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    Question inner product spaces

    Can you please, help me with this
    Suppose V and W are finite dimensional inner product spaces with orthogonal bases B1 and B2, respectively. Let T: V->W is linear, so we know that T*: V->W (linear) exists and is unique. Prove that [T*]_B1,B2 is the conjugate transpose of [T*]_B2,B1.
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  2. #2
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    Quote Originally Posted by bamby View Post
    Suppose V and W are finite dimensional inner product spaces with orthogonal bases B1 and B2, respectively. Let T: V->W is linear, so we know that T*: V->W (linear) exists and is unique. Prove that [T]_B1,B2 is the conjugate transpose of [T*]_B2,B1.
    Suppose that B_1 consists of vectors e_i, and B_2 consists of vectors f_j. Then \langle Te_i,f_j\rangle = \langle e_i,T^*f_j\rangle = \overline{\langle T^*f_j,e_i\rangle}. In other words, the (j,i)-element of [T^*]_{B_2,B_1} is the complex conjugate of the (i,j)-element of [T]_{B_1,B_2}.
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