Originally Posted by
HallsofIvy Finding bases for S and T is easy: if the given sets are independent, then they are bases. If not, then you can solve for one (or two) of the vectors in terms of the others.
For example, to see if <(1,0,1), (i,,-i,0), (0,i,i)> is an independent set, look at a(1,0,1)+ b(i,-i,0)+ c(0,i,i)= (0,0,0). That is the same as (a+ ib, -ib+ ic, a+ ic)= (0,0,0) so we have the three equations a+ ib= 0, -ib+ ic= 0, a+ ic= 0. The middle equation is the same as ib= ic or b= c. The first and second equations, both reduce to a= -ib. If, for example, we take b= 1, then a= -i and c= 1 so -i(1,0,1)+ (i, -i, 0)+ (0,i,i)= (-i+i, -i+i, -i+i)= (0,0,0). That set is not independent and we can now see that (i,-i,0)= i(1,0,1)- (0,i,i) so we don't need the vector (i,-i,0)- it can be written using the other two. So <1,0,1),(0,i,i)> is enough for a basis. (Actually, any two of the three vectors forms a basis.)