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Math Help - Affine transformation question...HELP!

  1. #1
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    Affine transformation question...HELP!

    In this question,
    f and g are both affine transformations. The
    transformation
    f is reflection in the line y = x 1, and the transformation
    g
    maps the points (0, 0), (1, 0) and (0, 1) to the points (3,1), (4,1) and
    (3,-2), respectively.
    (a) Determine
    g in the form g(x) = Ax + a, where A is a 22 matrix
    and
    a is a vector with two components.

    (b) Express
    f as a composite of three transformations: a translation,
    followed by a reflection in a line through the origin, followed by a
    translation. Hence determine
    f in the same form as you found g in
    part (a).

    I have answers to these questions but I'm really not sure if they're right, so if someone could put up the correct answers so I can see how I did then that would be great! Thanks



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  2. #2
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    Re: Affine transformation question...HELP!

    Quote Originally Posted by alexlbrown59 View Post
    In this question,
    f and g are both affine transformations. The
    transformation
    f is reflection in the line y = x − 1, and the transformation
    g
    maps the points (0, 0), (1, 0) and (0, 1) to the points (3,−1), (4,−1) and
    (3,-2), respectively.
    (a) Determine
    g in the form g(x) = Ax + a, where A is a 22 matrix
    and
    a is a vector with two components.

    (b) Express
    f as a composite of three transformations: a translation,
    followed by a reflection in a line through the origin, followed by a
    translation. Hence determine
    f in the same form as you found g in
    part (a).
    a)

    A(0,0) + a=(3,-1) so clearly

    $a=\left(\begin{array}{c}3\\-1\end{array}\right)$

    A(1,0) + (3,-1) = (4,-1)
    A(1,0) = (1,0)

    A(0,1) + (3,-1) = (3,-2)
    A(0,1) = (0,-1)

    $A =\left(\begin{array}{cc}1&0\\0 &-1\end{array}\right)$

    b)

    $t=\left(\begin{array}{c}0\\-1\end{array}\right)$

    $r=\left(\begin{array}{cc}0 &1\\1 &0\end{array}\right)$

    $f(x)=r(x-t)+t$
    Thanks from mash
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    Re: Affine transformation question...HELP!

    First, thanks for replying!
    My answer to part a is definitely right, and for part b I get (in the form (a,b,c,d) for the matrix)

    f(x) = (0,1,1,0)X + (1,-1)

    Please could you tell me if that's right too, otherwise my answers to the next part of the question will be wrong?!
    Last edited by alexlbrown59; March 21st 2014 at 01:40 AM.
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    Re: Affine transformation question...HELP!

    Quote Originally Posted by alexlbrown59 View Post
    First, thanks for replying!
    My answer to part a is definitely right, and for part b I get (in the form (a,b,c,d) for the matrix)

    f(x) = (0,1,1,0)X + (1,-1)

    Please could you tell me if that's right too, otherwise my answers to the next part of the question will be wrong?!
    I posted the correct form for f(x). What you have is incorrect. You neglect the first translation.
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    Re: Affine transformation question...HELP!

    Ok I'm wrong. When you take my expression and expand it out it results in what you have.
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    Re: Affine transformation question...HELP!

    Oh ok great, thanks! Then the question asks me to find the affine transformation g o f in the same form as I found g in part (a).
    I wrote if f(X) = BX+b and g(X) = CX+c then
    g o f = g(fX) = C(fX)+c = C(BX+b)+c = CBX+Cb+c = (CB)X+(Cb+c)

    Is this the right formula to use?
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    Re: Affine transformation question...HELP!

    Quote Originally Posted by alexlbrown59 View Post
    Oh ok great, thanks! Then the question asks me to find the affine transformation g o f in the same form as I found g in part (a).
    I wrote if f(X) = BX+b and g(X) = CX+c then
    g o f = g(fX) = C(fX)+c = C(BX+b)+c = CBX+Cb+c = (CB)X+(Cb+c)

    Is this the right formula to use?
    That looks correct. The "matrix" part will be CB, and the "translation part" will be translation by Cb + c, where b is the translation part of f and c is the translation part of g, and C is the matrix associated with g.
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  8. #8
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    Re: Affine transformation question...HELP!

    That's great, thank you!
    So I now have the affine transformation g o f which I found to be (0,1,-1,0)X + (4,0).
    The last part of the question asks
    Use your affine transformation g o f to show that there is exactly one point
    (x, y) such that the image of (x, y) under g o f is (x, y). State the
    coordinates of this point.
    If you could please just help me with this last part I would be so grateful
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    Re: Affine transformation question...HELP!

    Quote Originally Posted by alexlbrown59 View Post
    That's great, thank you!
    So I now have the affine transformation g o f which I found to be (0,1,-1,0)X + (4,0).
    The last part of the question asks
    Use your affine transformation g o f to show that there is exactly one point
    (x, y) such that the image of (x, y) under g o f is (x, y). State the
    coordinates of this point.
    If you could please just help me with this last part I would be so grateful
    you can manage this. Just solve for

    $v=Av+b$ where

    $v=\left(\begin{array}{r}x \\ y\end{array}\right)$, $A=\left(\begin{array}{rr} 0 &1 \\ -1 &0 \end{array}\right)$, and $b=\left(\begin{array}{r} 4\\ 0 \end{array}\right)$
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    Re: Affine transformation question...HELP!

    Thank you so much for all your help on this question. I actually understood it once you explained, just think I needed that initial start. My final answer is x=2 and y=-2, which, when I draw the diagram, seems to be correct. Thanks once again
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