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Math Help - Projection - linear transformation

  1. #1
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    Projection - linear transformation

    V = V1 \oplus V2 is a direct sum of two subspaces V1 and V2.

    T is the projection transformation of V onto V1, parallel to V2. Find ImT and KerT.

    This is a classic question and the answer is ImT = V1 and kerT = V2

    But, as usual, I'm having trouble understanding why. Can someone tell me if this is correct:

    since V = V1 \oplus V2, for every v \in V, v = v1 + v2.

    T(v) = T(v1+v2) = v1, so ImT = v1

    kerT is when the image is equal to 0 -- T(v) = T(v1+v2)=v1 -- but v1 is equal to zero, so T(v2) = 0.

    which makes kerT = V2.

    Is this the correct way to look at it??
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  2. #2
    Senior Member jakncoke's Avatar
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    Yes, correct. Ker(T) consists of elements of the form (0, p)  p \in V_2 . It is true that {0}  \oplus V_2 is isomorphic to  V_2 .
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