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A question on linear maps

  1. Aug 25, 2013 #1
    Suppose that V and W are finite dimensional and that U is a subspace of V. If dimU≥dimV-dimW prove that there exists a linear map T from V to W such that kerT=U.

    My answer is this:

    Consider the following linear map:
    T|u>=|u> if |u> belongs to V-U and T|u>=|0> if |u> belongs to U​

    Therefore kerT=U
    Is this correct?
     
  2. jcsd
  3. Aug 25, 2013 #2

    tiny-tim

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    hi kostas230! :smile:
    nooo …

    i] that's a map from V to V, not to W
    ii] what do you mean by V-U ? :confused: (eg what is R3 - R2 ?)
     
  4. Aug 25, 2013 #3
    Silly me, I overlooked it xD

    I mean the elements of V that do not belong in U.
     
  5. Aug 25, 2013 #4

    tiny-tim

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    then R3 - R2 would include all elements with z ≠ 0 :confused:
     
  6. Aug 25, 2013 #5
  7. Aug 25, 2013 #6

    tiny-tim

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    that's a line

    add a line to R2 and you don't get R3 :redface:
     
  8. Aug 25, 2013 #7

    HallsofIvy

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    R2 direct sum a line gives R3 but that is not at all what you said!


    R3- R2 is all (x, y, z) such that [itex]z\ne 0[/itex] which is not a subspace.
    I think you need to review your definitions!
     
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