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**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.)