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Linear operator in R2

  1. Mar 16, 2013 #1
    1. The problem statement, all variables and given/known data

    Let a be a fixed nonzero vector in R2. A mapping of the form L(x) = x + a is called a
    translation. Show that a translation is not a linear transformation. Illustrate geometrically the effect of a translation.

    My work is in the photo below, can you check and see if i'm correct.

    Also to show geometrically the effect of a translation, I just drew a vector being shifted a units.

    Thanks

    Higgenz



    2. Relevant equations



    3. The attempt at a solution

    259hcsw.jpg
     
  2. jcsd
  3. Mar 16, 2013 #2

    Fredrik

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    Staff Emeritus
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    Gold Member

    You should try to type your posts instead of posting a picture. Typed text is so much easier to read. If you want to type a matrix, you can do it like this:
    $$\begin{bmatrix}a & b\\ c & d\end{bmatrix}$$ Hit the quote button next to my post to see how I did this. More information here.

    The statement of the problem is a bit careless. There is a translation that's linear: the one with a=0. The problem should be asking you to prove one of the following statements:

    1. If L is a translation by ##a## and ##a\neq 0##, then L is not linear.
    2. If L is a translation by ##a## and L is linear, then a=0.

    The calculation you're doing is fine, but since you're just showing the calculation, and not including any "for all" or "there exists" statements, it's not clear what you're proving. If you decide to prove statement 1 above, you need to think about what exactly "L is not linear" means.

    Also, I think you should change the notation so that the components of u are denoted by ##u_1## and ##u_2##, and similarly for the other variables.
     
    Last edited: Mar 16, 2013
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