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Math Help - Please check for me, affine transformations

  1. #1
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    Please check for me, affine transformations

    Brief solutions at the bottom

    In this question, f and g are both affine transformations. The transformations f is a reflection in the line y = -x +1, and g maps the points (0,0), (1,0) and (0,1) to the points (1,5), (1,-4) and (0,-5) respectively.

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

    b) Write down the matrix that represents reflection in an appropriate line through the origin, and find f (in the same form as for g in part (a)) by first translating an appropriate point to the origin.

    c) Find the affine transformation g o f (in the same form as for g and f in parts (a) and (b)).

    d) Hence or otherwise, find the images of the points (0,0), (0,-2) (2,-2) and (2,0) under g o f. Mark these points and images on the same diagram, making it clear which points maps to which. Describe g o f geometrically as a single transformation.


    Here is what I got;



    a) g(x) = (0 -1)x + (1)
    1 0 -5

    b) f(x) = (0 -1) + (1)
    -1 0 1

    c) gof = (1 0)x + ( 0)
    0 -1 -6

    d) Using solution from (c) I got images are (0,-6) , (0, -4), (2,-4) and (2,-6)
    Otherwise I got images as (0,-4) , (0,-2) , ( 2,-2) and (2,-4)

    Donít know which if any is correct. I might have made mistakes in earlier workings.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by fair_lady0072002
    Brief solutions at the bottom

    In this question, f and g are both affine transformations. The transformations f is a reflection in the line y = -x +1, and g maps the points (0,0), (1,0) and (0,1) to the points (1,5), (1,-4) and (0,-5) respectively.

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

    b) Write down the matrix that represents reflection in an appropriate line through the origin, and find f (in the same form as for g in part (a)) by first translating an appropriate point to the origin.

    c) Find the affine transformation g o f (in the same form as for g and f in parts (a) and (b)).

    d) Hence or otherwise, find the images of the points (0,0), (0,-2) (2,-2) and (2,0) under g o f. Mark these points and images on the same diagram, making it clear which points maps to which. Describe g o f geometrically as a single transformation.


    Here is what I got;



    a)...........g(x) = (0 -1)x + (1.)
    .......................(1 0) .....(-5)
    if x=[0,0]', your g(x)=[1,-5]' rather than [1,5]' as is given in the question.

    RonL
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