Linear Transformation - Linear Algebra

  1. [SOLVED] Linear Transformation - Linear Algebra

    1. The problem statement, all variables and given/known data

    Determine if T is linear. T(x,y,z) = (1,1)

    2. Relevant equations

    Definition of Linear Transformation: A function T: R^n --> R^m is a linear transformation if for all u and v in R^n and all scalars c, the following principles are satisfied:

    Homogeneity Principle: T(cu) = cT(u)
    Additivity Principle: T(u+v) = T(u) + T(v)

    3. The attempt at a solution

    The answer that was given for this question is false, and I am trying to see why. Therefore, this is what I've done but I am missing something because I always get true if I am to assume that T(cx, cy, cz) = (c,c) but the answer is supposed to be (1,1).

    My reason thus far has been to say that because there are no variables in the solution of the transformation (1,1), then a scalar cannot be multiplied to it, but that doesn't make sense because it should work for all vectors, regardless of whether they are numbers or letters. The answer my prof gave was this:

    T(cx, cy, cz) = (1,1) which does not equal c(1,1) = cT(x,y,z). He didn't bother with the addivitivity since both principles need to be upheld for the transformation to be linear.

    If you could clarify, I would really appreciate it!
  2. jcsd
  3. EnumaElish

    EnumaElish 2,481
    Science Advisor
    Homework Helper

    1. Write out T(cu)
    2. Write out T(u)
    3. Multiply "2" with c
    4. Compare "3" with "1"
  4. Constant transformations should send up red flags. Remember that linear transformations form a vector space. So they need to work with the kind of manipulations EnumaElish described.
  5. I see it now! Thanks so much!
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