Hi everyone!

Iβm studying machine learning and came across a question that might seem simple, but itβs causing some confusion. I want to better understand the difference between the concept of linearity in algebra and statistical linearity, especially when it comes to regression. Algebraic Linearity: As far as I understand, algebraic linearity means that a function π¦ = π ( π₯ ) y=f(x) is linear if π₯ x appears only in the first degree, for example, π¦ = π π₯ + π y=mx+b. If we add terms like π₯ 2 x 2 or higher powers, the function becomes nonlinear. Statistical Linearity: On the other hand, in statistics, Iβve come across the claim that even if we have terms like π₯ 2 x 2 , π₯ 3 x 3 , and so on, the model can still be considered linear if itβs linear in the parameters (e.g., π¦ = π½ 0 + π½ 1 π₯ + π½ 2 π₯ 2 + π½ 3 π₯ 3 y=Ξ² 0 β +Ξ² 1 β x+Ξ² 2 β x 2 +Ξ² 3 β x 3 ). In this case, linearity is defined as linearity in the coefficients π½ Ξ². Question: Could someone explain in detail why the term βlinearityβ is used this way in statistics? Whatβs the reasoning behind this, and what advantages does it provide for data analysis? It would be helpful to see examples where understanding these differences is important in practice. Thanks!