Example: let v be the set of all pairs (x, y) of real number, and let f be the field of real no. define (x, y) + (x, y,) = (3y + 3y,, -x - x) c(x, y) = (3cy, cx

Example: Let V be the set of all pairs (x, y) of real number, and let F be the field of real no. define (x, y) + (x, y,) = (3y + 3y,, -x - x) c(x, y) = (3cy, cx) S Verify that V, with these operations is not a vector space over the field of real number. Solution: Take a (1, 2), B = (2, 3), y = (1, 3) Then (a +B) +Y = {(1, 2) + (2, 3)} + (1, 3) = (15, -3)+ (1, 3) = (0, -16) a+(B + y) = (1, 2) +{(2, 3) + (1, 3)} = (1, 2) + (18, -3) =(-3, -19) Thus we conclude that in general (a+)+Y a + (+Y)