This may have been asked before but I couldn't find it from a search.
I know that with simple linear regression, the regression slope is equivalent to Pearson's correlation coefficient $\rho$ for standardized variables [1]. Moreover, using the property of bilinearity of covariance, I know that the correlation coefficient can be used to express the correlation between linear combinations of variables [2].
When running a multiple regression between a dependent variable and a set of standardized independent variables, can it be shown that the fitting procedure of a multiple regression will give a linear combination $a_1X_1 + .. + a_nX_n$ of the independent variables that maximize $\rho(Y, a_1X_1 + .. + a_nX_n)$ where $a_1,..,a_n$ are the regression coefficients as determined by the multiple regression and Y is the dependent variable?
[1] What's the difference between correlation and simple linear regression?
