1. The problem statement, all variables and given/known data When are the vectors (1,x) and (1,y) linearly independent in R^2? When are the vectors (1,x,x^2), (1,y,y^2), and (1,z,z^2) linearly independent? Generalize to R^n. 3. The attempt at a solution At first, I think I misconstrued the question and I ended up finding basis vectors. ie, this is what I did for R^2: In order for the vectors (1,x) and (1,y) to be linearly independent, there must exist some c1,c2 such that c1(1,x)+ c2(1,y)= (0,0) then we have c1=-c2 c1x=-c2y Using the first equation you have C=(c1,c2)=c2(-1,1) and so the basis vector is (-1,1). I don't really know why i did this honestly. I just saw linearly independent, and from there I got to my last step. Then upon actually reading the question again, I realized that for them to be linearly independent their dot product should be 0. Also I guess my initial way of solving the problem was wrong since I should have gotten that c1=c2=0, right? So anyway if the dot product=0 then <(1,x),(1,y)> = 1+xy and we want this equal to 0 --> xy=-1. Any help?