es
   
Example 1: Let us consider polar coordinates on the plane ℝ²: \[ x = r\,\cos\theta , \quad y = r\,\sin\theta , \] or in complex plane ℂ: \[ z = r\,e^{{\bf j}\,\theta} , \] where is the imaginary unit on the complex plane, so j² = −1. So every point (x, y) on the plane ℝ² is identified by the pair of real numbers (r, θ). However, this transformation from Cartersian coordinates (x, y) to polar coordinates (r, θ) is not linear because \[ \left( r_1 , \theta_1 \right) + \left( r_2 , \theta_2 \right) \ne \left( r_1 + r_2 , \theta_1 + \theta_2 \right) . \] However, this transition to polar coordinates in bulinear because \[ \left( r_1 , \theta \right) + \left( r_2 , \theta \right) \ne \left( r_1 + r_2 , \theta \right) .\quad\mbox{and} \quad \left( r , \theta_1 \right) + \left( r , \theta_2 \right) \ne \left( r , \theta_1 + \theta_2 \right) . \]    ■
End of Example 1
   
Example 2:    ■
End of Example 2

 

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