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Linear transformation proof?

  1. Mar 14, 2013 #1
    1. The problem statement, all variables and given/known data

    See attached image below.

    2. Relevant equations



    3. The attempt at a solution

    I know for it to be a linear transformation it must be that: f(x)+f(y)=f(x+y) and f(tx)=tf(x) where t is a scalar. I'm not sure where to start with this proof.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     

    Attached Files:

  2. jcsd
  3. Mar 14, 2013 #2

    Mark44

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    It's not very clear from the description but the transformation could also be "desribed" this way: f : R → M2x2,
    where
    $$ f(c) = \begin{bmatrix} 1 & c \\ 0 & 1\end{bmatrix}$$

    Start by calculating f(x), f(y), and f(x + y) and seeing if f(x) + f(y) = f(x + y). Then check that f(tx) = t*f(x).
     
  4. Mar 14, 2013 #3

    LCKurtz

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    Do you understand what the domain of the transformation is and its "formula" as it acts on the domain?
     
  5. Mar 14, 2013 #4

    LCKurtz

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    How do you know it isn't a transformation from ##R^2## to ##R^2##?
     
  6. Mar 14, 2013 #5

    Fredrik

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    You didn't actually post the definition of ##f_A##. The image says that ##f_A## is "described" by that matrix, but what does that mean?


    I don't think this is right. The problem said we should check that ##f_A## is linear for all c. So, there should be one function for each value of c. The function you called f is just one function. It's more likely that we should check that what you called f(c) is linear, i.e. that for all real numbers c, we have f(c)(ax+by)=af(c)x+bf(c)y for all vectors x,y and all real numbers a,b.

    The OP should explain how ##f_A## is defined.
     
  7. Mar 14, 2013 #6
    I'm not sure. We only did a little on matrix transformations. Would f(x) be that matrix with x instead of c?
     
  8. Mar 14, 2013 #7

    Mark44

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    What I wrote is how I interpreted the OP's attachment.
    Yes. It was unclear to me, as well.
     
  9. Mar 14, 2013 #8
    This is the original question, as stated on the handout.
     

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  10. Mar 14, 2013 #9

    Fredrik

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    OK, but the notation ##f_A## where A is a matrix, or the concept of a function being "described" by a matrix, should be explained somewhere in the book.

    I would guess that you're supposed to show that regardless of the value of c, the function ##f_A:\mathbb R^2\to\mathbb R^2## defined by ##f_A(x)=Ax## for all ##x\in\mathbb R^2##, is linear. (It is however a little bit weird to ask for this, since these functions are linear for all 2×2 matrices A, not just the ones mentioned in the problem. So you should still try to find a definition in your book or in some other handout).
     
  11. Mar 14, 2013 #10

    Fredrik

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    The statement of problem 8 (which you posted here) strongly suggests that ##f_A## is to be interpreted the way I did in my previous post.
     
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