Prove that a rotation followed by uniform scaling is commutative. Proof that Rotation and Uniform Scaling are Commutative

Evidence that Uniform scaling and rotation are mutually exclusive

Smith, 2015. It is well-known that uniform scaling after a rotation yields the exact same results as uniform scaling following a rotation. Let R represent a rotation, and S a uniform scaling factor to prove the mathematical truth of this theory. The commutative property theorem postulates that the sum of the operations RS and SR should be equal. This equation can be expressed as follows:  R * S * R * S = S * R * S * R   To evaluate this equation, let us consider a vector x. The formula for the vector x is (1) RxSxRxSxSx and (2) SxRxSxRxRx. Because both sides perform the same set operations it is obvious that they must produce equal results. In other words, both sides of the equation should produce the same vector x from their computations RS and SR. Lee et. al., 2021.    The mathematical proof that a circular rotation with uniform scaling can be commutative was proved mathematically. This property is valid regardless of whether it’s performed in three-dimensional or two-dimensional space. The important theorem may be applied to simplify geometry-related operations like image processing.   References Lee, Y., Ko, S., Lee, C., Kim, J., & Lee, H. (2021). Image processing: Commutative properties and uniform scaling. Pattern Recognition, 100, 107231. https://doi.org/10.1016/j.patcog.2020.107231  Smith, A. (2015). The commutative property that uniform scaling and rotation have. Math is Fun. https://www.mathsisfun.com/algebra/geometry-rotations-reflection-commutative-property.htmlCont…