I've been noticing that statisticians might be using the term "linear" in a bit sloppy way.
A linear model is a model of the form:
$$Y = \beta_1X_1+\beta_2 X_2 + \cdot \cdot \cdot+ \beta_nX_n$$
Which in mathematical terms, is a linear combination (betas are scalars, $X$s are vectors).
However, statisticians allow a linear model to have other than $\le 1$ order polynomial terms. Such as quadratic terms $X_i^2$. In mathematics, polynomials of order $> 1$ are not linear functions. To me this sounds like the statistics concept of a linear model is a bit sloppy.
What do you think of the terms and how do you interpret "linearity" in linear models? Is it the same as in mathematics?