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Question about linear transformations

  1. Jul 23, 2014 #1
    1. The problem statement, all variables and given/known data
    Hey PF, I'm here again asking about linear transformations, ha ha.
    Let C={(x,y) [itex]\in[/itex] [itex]\mathbb R[/itex]2 | x²+y²≤1} a circle of radius 1 and consider the linear transform
    T:[itex]\mathbb R[/itex]2→[itex]\mathbb R[/itex]2
    (x,y) [itex]\mapsto[/itex] ([itex]\frac{5x+3y}{4}[/itex],[itex]\frac{3x+5y}{4}[/itex])​
    Find all values of a natural n for which Tn(C), the image of C after n applications of T, contains at least 2013 points (a,b) with coordinates a, b [itex]\in[/itex] [itex]\mathbb Z[/itex].
    2. Relevant equations
    N/A


    3. The attempt at a solution
    At first I've fixed for both input and output basis for the map ε = {e1,e2} (i.e. the canonic basis) writing the linear transformation in a matrix form:
    [tex]\begin{pmatrix}
    T\end{pmatrix}^{ε}_{ε}\begin{pmatrix}
    x\\
    y
    \end{pmatrix}_{ε}
    =\begin{pmatrix}
    \frac{5}{4} & \frac{3}{4}\\
    \frac{3}{4} & \frac{5}{4}
    \end{pmatrix}\begin{pmatrix}
    x\\
    y
    \end{pmatrix}_ε[/tex]
    Then, knowing I'm going to be applying the linear transformation n times, I thought would be a wise choice to diagonalize it, taking advantage of it's nice form to do so:
    [tex]\begin{pmatrix}
    T\end{pmatrix}^{ε}_{ε} = -\frac{1}{2}\begin{pmatrix}
    1 & 1\\
    1 & -1
    \end{pmatrix}\begin{pmatrix}
    2 & 0\\
    0 & \frac{1}{2}
    \end{pmatrix}\begin{pmatrix}
    -1 & -1\\
    -1 & 1
    \end{pmatrix}[/tex]
    But right now, I'm having a little bit of trouble to figure out how to count the number of points on the maps that has both coordinates belonging to [itex]\mathbb Z[/itex]. Hope for some help, thanks in advance guys.
     
  2. jcsd
  3. Jul 23, 2014 #2
    I will respond, but I am not sure how much help I can offer. I have two questions, at least.

    1. What does the image of a circle under a linear map look like?

    2. We have spoken before about how the the determinant of the matrix associated with a linear map will tell you how area scales. So what is then the area of the transformed region after ##n## applications of the mapping?

    After answering these questions, do you have any thoughts on it?
     
  4. Jul 23, 2014 #3
    Well, I can see that after you apply the transformation on the circle, we expand vectors in the (1,1) by a factor of 2, and contract the ones in the (1,-1) direction by a factor of two either. Then, the image of the circle after applying T n times is limited by an ellipse centered at the origin, with semi-axis sized 2^n, in the (1,1) direction, and 1/(2^n), in the (1,-1) direction.
    Since det(T)=1, applying the transformation doesn't actually change the value of the area.
    My thoughts on it would be to count only the (c,c); c [itex]\in[/itex] [itex]\mathbb Z[/itex], since I'm sure that the number of those kind of points is growing with each time I apply T in the region C. Is this a good way to go?
     
  5. Jul 23, 2014 #4
    Right. It would seem to me as well that as ##n## grows larger the image looks more and more like the line ##y=x##.

    Maybe you can use your diagonalization to determine for which ##n## we reach points like ##(1000,1000)##. You have ##T^n \mathbf{x} = P^{-1}D^nP \mathbf{x}##, right?

    Do you have access to an answer to the question?
     
  6. Jul 23, 2014 #5
    Hm, I'm gonna try that indeed... I guess the right answer would be n≥11. I think I would be able to find somewhere an answer model, since it was a question in a Math Olympiad. If you are interested let me now.
     
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