Understand and use multiplication by a complex number as a linear transformation in the complex plane.

The table below shows four complex numbers and their images under a linear transformation T defined by T(z) = wz, where w is a fixed complex number. Determine the values of p, q, and r, and show that w = √2 cis(5π/12).

Consider the complex number \(w = 1 + i\). a) Determine the modulus and argument of \(w\), expressing the argument in exact form. b) The complex number \(z = 3 - i\) is multiplied by \(w\) to give \(z' = wz\). Use your results from part (a) to describe the geometric transformation that maps \(z\) to \(z'\) in the complex plane.

Consider the complex number \(w = \sqrt{3} + i\). a) Express \(w\) in modulus-argument form. (1 mark) b) The transformation \(T\) maps a complex number \(z\) to \(wz\). Use your result from part (a) to describe geometrically the effect of transformation \(T\) on any complex number \(z\) in the complex plane. (2 marks)

The complex number \(z = 2 + 3i\) is multiplied by \(w = \operatorname{cis}\left(\frac{\pi}{3}\right)\). Determine the modulus of the resulting complex number \(zw\).