The problem you provided asks to show that the map $\rho_{\zeta} : \mathbb{R}^2 \to \mathbb{R}^2$, defined by $\rho_{\zeta}(x) = \zeta \cdot x$, represents the rotation of the complex plane by the angle $\theta$, where $\zeta = e^{i\theta} = \cos(\theta) + i \sin(\theta)$.

Solution Outline:

1. Interpretation of the map $\rho_{\zeta}$:

   • The complex number $\zeta = e^{i\theta}$ represents a point on the unit circle in the complex plane. Multiplication by $\zeta$ rotates any complex number by the angle $\theta$.
   • Given $x \in \mathbb{R}^2$, think of $x$ as a vector in the complex plane, identified as $x_1 + ix_2$, where $x = (x_1, x_2)$.

2. Multiplying by $\zeta$:

   • To rotate a vector $x = x_1 + ix_2$ by an angle $\theta$, we multiply it by $\zeta = \cos(\theta) + i \sin(\theta)$: $\zeta \cdot x = (\cos(\theta) + i \sin(\theta))(x_1 + ix_2)$ Expanding this: $\zeta \cdot x = \cos(\theta)(x_1 + ix_2) + i \sin(\theta)(x_1 + ix_2)$ Simplifying this expression yields: $= \cos(\theta) x_1 - \sin(\theta) x_2 + i(\sin(\theta) x_1 + \cos(\theta) x_2)$ This can be separated into its real and imaginary parts: $= (\cos(\theta) x_1 - \sin(\theta) x_2) + i(\sin(\theta) x_1 + \cos(\theta) x_2)$ Therefore, after rotation by $\theta$, the new coordinates are: $(\cos(\theta) x_1 - \sin(\theta) x_2, \sin(\theta) x_1 + \cos(\theta) x_2)$ This is the standard form of a 2D rotation matrix applied to $(x_1, x_2)$: \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}$$ which confirms that multiplication by $$\zeta = e^{i\theta}$$ rotates the vector by $$\theta$$.

Conclusion:

Thus, we have shown that the map $\rho_{\zeta}(x) = \zeta \cdot x$ is indeed the rotation of the complex plane by the angle $\theta$.

Would you like more details on any part of this solution?

Here are five related questions you might explore next:

1. How does the rotation matrix change if the angle $\theta$ is negative?
2. What is the geometric interpretation of multiplying a complex number by $e^{-i\theta}$?
3. Can we extend this concept of rotation to 3D space? If so, how?
4. How would you prove the same result using polar coordinates instead of Cartesian?
5. What happens to a complex number when multiplied by $\zeta = e^{i\pi}$?

Tip: Complex number multiplication often leads to rotations and scalings in the plane, a key concept in both algebra and geometry.