I need help with this problem, I'm very confused
Prove that for all complex numbers z and w:
1. zw* + z*w is always real
2. zw* - z*w is purely imaginary or zero.
if you mean by z* the conjugate here is the answer
let
$\displaystyle z = x+iy , w = u+iv $
$\displaystyle \bar{z} = x -iy , \bar{w} = u-iv $
$\displaystyle z\bar{w} = xu+yv+i(yu-vx) $
$\displaystyle \bar{z} w = xu+vy +i(vx-uy) = xu+vy -i(uy-vx)$
$\displaystyle z\bar{w} + \bar{z} w = xu+yv+i(yu-vx)+xu+vy -i(uy-vx) = 2(xu+yv) $
and this is always real
$\displaystyle z\bar{w} - \bar{z} w = xu+yv+i(yu-vx) - (xu+vy -i(uy-vx)) = 2i(yu-vx) $
and this maybe 0 if yu=vx or imaginary if yu dose not equal vx
and it is known that
$\displaystyle z + \bar{z} = 2Re(z) $ Re is the real part of z
$\displaystyle z - \bar{z} = 2Im(z) $ Im is the imaginary part of z