Checking linearity

Suppose I want to see if Gaussian linear regression is an appropriate model for my data. Then let X be the design matrix (with each column corresponding to an explanatory variable). and let e be the vector containing my residuals. We then have transpose(X)e = 0.

My course says that this implies that "no correlation should appear between explanatory variables and residuals".

I do not understand what is meant by correlation here as the definition I know involves two random variables. But X is not a random variable.

Also, given the right definition, how does he arrive at this conclusion of "no correlation"?

Thank you in advance! I hope my question was adequate as this is my first time posting.

• The match between the title (linearity, correlation?), the body and the tags (linearity, correlation, assumptions?) could be improved. – Richard Hardy Nov 6 '19 at 18:48