The question:

Find the image of the line x - y = 2 in the complex plane under the mapping w = iz - 1.

My solution:

$\displaystyle z = \frac{w + 1}{i}$

Let z = x + iy, w = a + ib

$\displaystyle \frac{(a + 1) + ib}{i} . \frac{-i}{-i}$

b - i(a + 1)

Equating real and imaginary parts,

x = b

y = (a + 1)

Sub into equation of the line:

b + (a + 1) = 2

b = 1 - a

Is this correct? Thanks.