How does a linear transformation of the form w = az + b affect the mapping of straight lines from the z-plane to the w-plane?

A linear transformation of the form w = az + b maps straight lines in the z-plane to straight lines in the w-plane. This holds true as long as the transformation is linear, meaning both a and b can be real or complex numbers. The transformation may also involve translation, magnification, or rotation, but the resulting image will still be a straight line if the original is straight.

The transformation w = az + b, where a and b are constants (real or complex), ensures that straight lines in the z-plane correspond to straight lines in the w-plane. This is because linear transformations preserve the linearity of the relationships between points. For example, if a straight line in the z-plane is represented by a linear equation, its image in the w-plane will also be represented by a linear equation, thus maintaining the straight-line property throughout the transformation process.