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Math Help - Affine transformations

  1. #1
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    Affine transformations

    Hi, the complete original question is:

    f and g are both affine transformations. The transformation f is a reflection in the line x=2, and the transformation g maps the points (0,0), (1,0) and (0,1) to the points (0, -4), (0, -3) and (-1, -4) respectively.

    a) determine g (in the form g(x) Ax + a, where A is a 2x2 matrix and a is a vector with two components.

    b) Exprerss 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)).

    c) Fine the affine transformation g o f (in the same form as you found g in part a)).

    d) Given that g o f is either a rotation or a reflection, state which it is. If it is a rotation, state the centre and angle of rotation. If it is a reflection, state the axis of reflection. Justify your answer.

    For a) I found that (g = Bx + b) g(x) =
    \left(\begin{array}{cc}0&-1\\1&0\end{array}\right) + \left(\begin{array}{cc}0\\-4\end{array}\right)

    For b) I found that (f = Ax + a) f(x) =
    \left(\begin{array}{cc}-1&0\\0&1\end{array}\right) + \left(\begin{array}{cc}4\\0\end{array}\right)

    c) I am unsure how to combine the two answers above. I thought something like: (?)

    g o f = g(f(x))
    = g(Ax + a)
    = B(Ax + a) + b


    d) Also I am not sure how to work out this question.

    Any help would be much appreciated. Thanks for your help in advance.

    Kind regards,

    Cozza
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  2. #2
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    stuck too

    hi cozza
    did you get this
    im doing it as part of a OU tma
    stuck on part c aswell
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  3. #3
    Junior Member
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    This is what my tutor said:

    g(f(x)) = g(Ax + a)
    = B(Ax + a) + b

    You are absolutely right so far. Now you need to multiply out the bracket
    just as if you were working with algebra except here you will be
    multiplying matrices & vectors together. You will then end up with
    something in the form Cx + c.

    I can't point you to a similar example in the text as I know there isn't
    one!

    I have multiplied out the bracket (I hope it's right!) ansd got an answer in the form Cx + c, but I'm not sure about part d) now.

    Hope that helps. Let me know if you have any further luck
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  4. #4
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    i got for a: 0 -1 x + 0
    _________ 1 0_____-4

    then for b i have used reflection in y axis matrix: -1 0
    __________________________________________0 1

    and got ans b to be: -1 0 x + 0
    ___________________0 1____4

    then ans c i got: 1 0 x + -4
    _______________0-1____-4

    ?
    what are you using for reflection matrix?
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  5. #5
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    Hi for part (c) I understand and end up with this below,
    g o f = Cx + c
    C = 0 -1
    and c = 0
    -1 0 0
    You can calculate this by multiplying the matrix for g with the matrix for f, so
    C = A*B and c = A*b + a
    Its just the part (b) that is really throwing me now as how I got to here (part c)

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