I'm aware that linear transformations of individual features do not affect Random Forest.  

But what if features were linearly recombined to construct a new feature?

**I do the following:** 
I take each sample, i.e. each training example, and linearly combine the features for that training example.  The result of this combination is a single value, and this single value becomes another feature for this training example.  For all of the samples, I do the same:  Linearly combining all of the values for the features, creating this new feature. The resulting data matrix of course has an extra column for this extra feature.  

 - Does this impact the results of a Random Forest?     
      
 - It is not a linear/monotonic transformation of any one of the features, but rather it is a linear combination of all of the features for a single training example.