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    1. A transformation from the z-plane to the w-plane is given by
    w=(z-i)/z
    show that under this transformation the line Im(z)=1/2 is mapped to the circle with equation |w|=1
    Hence, or otherwise, find, in the form w=(az+b)/(cz+d) where a,b,c, and d are C(you know what i mean), the transformation that maps the line Im(z)=1/2 to the circle centre(3-i) and radius 2.


    2.
    M is the matrix
    101
    020
    431
    (a) Find the eigenvalues and corresponding eignvectors for the matrix M.( I've done it)
    (b)
    A transformation T:R^3->R^3 is represented by the matrix M. Find catrtesian equations of the image of the line with equations
    x/2=y=-z/1
    under this transformation.
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    (Original post by totaljj)
    1. A transformation from the z-plane to the w-plane is given by
    w=(z-i)/z
    show that under this transformation the line Im(z)=1/2 is mapped to the circle with equation |w|=1
    Hence, or otherwise, find, in the form w=(az+b)/(cz+d) where a,b,c, and d are C(you know what i mean), the transformation that maps the line Im(z)=1/2 to the circle centre(3-i) and radius 2.
    p + qi
    = (x - (1/2)i) / (x + (1/2)i)
    = (x - (1/2)i)^2 / [(x + (1/2)i)(x - (1/2)i)]
    = (x^2 - 1/4 + xi) / (x^2 + 1/4).

    So

    p = (x^2 - 1/4) / (x^2 + 1/4),
    q = x / (x^2 + 1/4).

    So

    p^2 + q^2
    = [(x^2 - 1/4)^2 + x^2] / (x^2 + 1/4)^2
    = (x^2 + 1/4)^2 / (x^2 + 1/4)^2
    = 1.

    For the second part, you need to simplify 2(z - i)/z + (3 - i).
    2.
    M is the matrix
    101
    020
    431
    (a) Find the eigenvalues and corresponding eignvectors for the matrix M.( I've done it)
    (b)
    A transformation T:R^3->R^3 is represented by the matrix M. Find catrtesian equations of the image of the line with equations
    x/2=y=-z/1
    under this transformation.
    Choose a point P other than the origin on the line. Find the equation of the line through O and T(P).
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    I dont understand why
    For the second part, you need to simplify 2(z - i)/z + (3 - i).
    could you explain for me thx
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    (Original post by totaljj)
    I dont understand why
    For the second part, you need to simplify 2(z - i)/z + (3 - i).
    could you explain for me thx
    w = (z - i)/z gives you a circle with centre O and radius 1.

    w = 2(z - i)/z gives you a circle with centre O and radius 2.

    w = 2(z - i)/z + (3 - i) gives you a circle with centre 3 - i and radius 2.
 
 
 
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