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Homework Help: Determine if the transformation is linear if T(x, y)= (x+1, 2y)

  1. Feb 5, 2010 #1
    1. The problem statement, all variables and given/known data

    Determine if the transformation T: [tex]R^2\rightarrow R^2[/tex] is linear if T(x, y)= (x+1, 2y)


    2. Relevant equations
    1. T(u + v) = T(u) + T(v)
    2. T(c*u) = cT(u)
    3. T(0) = 0

    3. The attempt at a solution

    (1). I'm not sure how to prove the first condition (additivity). Can anyone help?

    (2). T(c x,c y) = (c x+1, c 2y) = c(x+1, 2y) =c T(x,y)

    For some scalar c.

    (3). In (2) if c=0
    T(0 x,0 y) = (0 x+1, 0 2y) = 0(x+1, 2y) =0
     
  2. jcsd
  3. Feb 5, 2010 #2

    Mark44

    Staff: Mentor

    Re: Transformation

    You may or may not be able to prove any of these conditions. What you need to do is to verify that they are all true, or show that one or more are not true.

    For the first one, let u = (u1, u2) and v = (v1, v2)
    Is T(u + v) = T(u) + T(v)?
    In the previous equation, you are trying to show that T(cu) = cT(u), for any scalar c, and any vector u.

    If u = (x, y), as you are assuming, what is T(cx, cy)? What you have after that step is not correct.
    No, T(0) is not 0. Remember that here 0 is a vector - (0, 0). What does T do to (0, 0)?
     
  4. Feb 5, 2010 #3
    Re: Transformation

    So, for the first condition, T(u+v) = (u1 + v1+1, 2 u2+v2) is not equal to (u1, 2u2) + (v1+1, 2u1) = T(u)+T(v).

    For the third condition, yes it will be (1,0) noy (0,0).

    T(cx,cy)=(cx+1,2cy) = c(x+1,2y)= cT(x,y)

    Is it right?
     
    Last edited: Feb 5, 2010
  5. Feb 5, 2010 #4

    Mark44

    Staff: Mentor

    Re: Transformation

    So what do you conclude about this transformation?
     
  6. Feb 5, 2010 #5
    Re: Transformation

    It is not linear since it violates at least one of the conditions for linearity. Thanks!!
     
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