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Math Help - Linear projection problem

  1. #1
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    Exclamation Linear projection problem

    One more problem found - please help if you can:

    The linear projection f is set with (2,1) -> (2,1,1) (1,1) -> (1,-1,2). Find the matrix that fits this linear projection in relation to the standard base, his formula and his core (subspace of all domain elements that project in the zero of the codomain).

    tnx
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  2. #2
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    So we know that

    T(2e_1+e_2)=2e_1+e_2+e_3 \iff 2T(e_1)+T(e_2)=2e_1+e_2+e_3

    and

    T(e_1+e_2)=e_1-e_2+2e_3 \iff T(e_1)+T(e_2)=e_1-e_2+2e_3

    Now we have a system of equations for the transformation of the basis vectors

    So if we subtract the 2nd equation from the first we get

    T(e_1)=e_1+2e_2-e_3

    subbing into either equation gives

    T(e_2)=-3e_2+3e_3

    Now these are the columns of the transformation matrix.
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