What is the R squared of a regression where none of the variables are collinear? If a regression has 100 observations and 100 variables and none of them are collinear, what is the r squared? I know that it means that the full rank assumption is satisfied, so does this mean the r squared equals 1? Also, what would the t-statistic for the 5th coefficient be?

If none of the explanatory variables is colinear (i.e., each pair has zero correlation) then the coefficient-of-determination for the regresion is equal to the sum of the coefficients-of-determination for individual simple linear regessions of each explanatory variable against the response variable.  If you would like to learn more about the relationships between collinearity and the coefficient-of-determination in linear regression, as wel as a broader geometric view of regression, you can find some discussion of this topic in my paper O'Neill (2019).
