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Math Help - General Question about Image

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    General Question about Image

    I'm not quite sure if I'll butcher the wording of this question, but here goes...

    How would I prove that given a linear map T:V-->V, such that the composition T^2 = I, and also given P and Q s.t. P = (I+T)/2 and Q = (I-T)/2, that Ker P = Im Q, and Im P = Ker Q?

    I think I'm just a little fuzzy about the whole concept of how to prove that something is in the image of P or Q. I understand Ker P is when (I+T)/2 = 0, and when I solve for T and plugin to Q, I get Q = I. Does this mean that it is in the image of Q? Or am I missing something here? .

    Also, as an aside, what would P and Q be called in this question? Not a "composition," right?

    Thanks in advance!
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  2. #2
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    Quote Originally Posted by limddavid View Post
    I'm not quite sure if I'll butcher the wording of this question, but here goes...

    How would I prove that given a linear map T:V-->V, such that the composition T^2 = I, and also given P and Q s.t. P = (I+T)/2 and Q = (I-T)/2, that Ker P = Im Q, and Im P = Ker Q?

    I think I'm just a little fuzzy about the whole concept of how to prove that something is in the image of P or Q. I understand Ker P is when (I+T)/2 = 0, and when I solve for T and plugin to Q, I get Q = I. Does this mean that it is in the image of Q? Or am I missing something here? .

    Also, as an aside, what would P and Q be called in this question? Not a "composition," right?

    Thanks in advance!
    First notice that for any v \in V

    T^2(v)=T(T(v))=I(v)=v and

    P(v)=\frac{1}{2}I(v)+\frac{1}{2}T(v)

    Q(v)=\frac{1}{2}I(v)-\frac{1}{2}T(v)

    PQ(v)=P(\frac{1}{2}I(v)-\frac{1}{2}T(v))=\frac{1}{2}P(I(v))-\frac{1}{2}P(T(v))=

    \frac{1}{4}I(I(v))-\frac{1}{4}T(I(v))+\frac{1}{4}I(T(v))-\frac{1}{4}T(T(v))

    \frac{1}{4}v-\frac{1}{4}T(v)+\frac{1}{4}T(v)-\frac{1}{4}T^2(v))

    \frac{1}{4}v-\frac{1}{4}T(v)+\frac{1}{4}T(v)-\frac{1}{4}v=0

    A similar computation shows

    QP(v)=0

    To show two sets are equal A=B

    Show that

    A \subset B and B \subset A

    let v \inKer (P) then P(v)=0 and

    Q(P(v))=Q(0)=0 \in Im (Q)

    This gives Ker (P) \subset Im (Q)

    Let w \in IM (Q) then w=Q(v) for some v \in V

    PQ(v)=0=P(w) \implies w \in Ker (P)

    So the two sets are equal.

    This should get you started.
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