Consider the vector space R^2 with the inner product given by:
<(x1,x2),(y1,y2)>=2x1y1-x2y1-x1y1+8x2y2.(note: don't verify this is an inner product)
Given the functional fi:R^2 --> R, fi(x1,x2)=6x1-x2, we know that there is a unique vector v=(y1,y2) such that fi(u)=<u,v> for every u=(x1,x2) in R^2.
Find the vector v. (recall you can verify the correctness if you want)
Now this is what I have done but i'm not sure if its right:
Since (x1,x2)=6x1-x2 in R, then : <(x1,x2),(6,-1)>=6x1-x2 therefore our
v=(6,-1) in R, now I want to find v in R^2 so: <(x1,x2),(y1,y2)>=x1y1+x2y2 = 2x1y1-x2y1-x1y2+8x2y2
then 0=x1y1-x2y1-x1y2+7x2y2= x1(y1-y2)+x2(-y1+7y2)
<(x1,x2),(y1-y2,-y1+7y2)>=(plug in 6,-1: (6-(-1),-6+7(-1))=(7,-13) in R^2. done
Is this correct? any help would be great. thank you!