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Linear Transformations Rn->Rm Question

  1. Apr 20, 2005 #1
    Linear Transformations Rn-->Rm Question

    I would be very grateful if someone can explain what is going on in the following problem:

    Determine whether the following T:Rn to Rm

    T(x,y)=(2x,y)

    Solution from solutions manual:

    T((x1,y1) + (x2,y2)) = (2(x1+x2), y1+y2) = (2x1,y1) + (2x2,y2) = T(x1,y1) + T(x2,y2)

    My questions are

    1. Where did the x1's and the x2's and the y1's and the y2's come from?

    2. Can you please explain whats happening step by step?



    [PS]--> The questions asks to use the theorem which states:

    A transformation T:Rn --> Rm is linear if and only if the following relationships hold for all vectors u and v in Rn and every scalar c

    a) T(u+v) = T(u) + T(v)

    b)T(cu) = cT(u)
     
  2. jcsd
  3. Apr 20, 2005 #2

    quasar987

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    He has set [itex]\vec{u} = (x_1,y_1), \ \ \vec{v} = (x_2,y_2)[/itex] and showed using vector addition properties that [itex]T(\vec{u}+\vec{v}) = T(\vec{u})+T(\vec{v})[/itex]
    This proof is imcomplete though because he left out condition b).
     
  4. Apr 20, 2005 #3
    Thank you quasar987 for the clarification. The manual does include the proof using condition b) but I forgot to type it.

    Thanks for that clarification.
     
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