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Show that T is a nonlinear transformation

  1. May 9, 2016 #1
    1. Show that T isn't a linear transformation and provide a suitable counterexample.
    ##T \begin{bmatrix}x\\y \end{bmatrix} = \begin{bmatrix}x - 1 \\ y + 1 \end{bmatrix}##

    2. The attempt at a solution
    ##\text{let}\, \vec{v} = \begin{bmatrix}0\\0 \end{bmatrix}. \text{Then,}##
    ##T(\vec{v}) = T\left(\begin{bmatrix}0\\0 \end{bmatrix}\right) = \begin{bmatrix}0 - 1 \\ 0 + 1 \end{bmatrix} = \begin{bmatrix}-1 \\ 1 \end{bmatrix}##
    Nonlinear transformation due to constants: ##T \begin{bmatrix}-1 \\ 1 \end{bmatrix}##.
     
  2. jcsd
  3. May 9, 2016 #2

    Mark44

    Staff: Mentor

    Why does the above show that T is not a linear transformation?
    And why is ## \begin{bmatrix}0\\0 \end{bmatrix}## a counter example?
    Your reason doesn't make sense to me, and this part -- ##T \begin{bmatrix}-1 \\ 1 \end{bmatrix}## -- really doesn't make sense.

    What are the properties a linear transformation has to satisfy?
     
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