Give an example of a quadratic form q(x; y) such that there are two vectors u= (x_u; y_u), v= (x_v; y_v) with q(u) = 0 and q(v) = 0 but q(u+v) does not= 0 Solution q(x,y) = (x-2y)*(x-y) Roots are x=2y , x=y Say (2,1) & (1,1) add them we get (3,2). this is not a root of q(x,y). Hence the example.

1. Give an example of a quadratic form q(x; y) such that there are two vectors u= (x_u; y_u), v=
(x_v; y_v) with q(u) = 0 and q(v) = 0 but q(u+v) does not= 0
Solution
q(x,y) = (x-2y)*(x-y)
Roots are x=2y , x=y
Say (2,1) & (1,1) add them we get (3,2). this is not a root of q(x,y). Hence the example