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Math Help - question about projection

  1. #1
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    question about projection

    Dears,
    I need the solution of the following exercise:

    Let V be a real vector space and E an idempotent linear operator on V, i.e.,
    a projection. Prove that (I + E) is invertible. Find (I + E)-1. (where I is the identity operator)

    Best Regards
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  2. #2
    MHF Contributor

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    Tejas
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    Re: question about projection

    hint:

    suppose A is a matrix such that (I + E)A = I.

    then A + EA = I.

    applying E to both sides we have:

    E(A + EA) = E

    EA + E2A = E

    but E2 = E (since E is a projection) thus:

    2EA = E, or EA = (1/2)E.

    now substitute this back in the equation

    A + EA = I, and solve for A.
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