Let V = R|. For u, v V| and a R| define vector addition ny u v: = u+v+3| and scalar multiplication by a u: = au+3|. It can be shown that (V,,)| is a vector space over the scalar field R|. Find the following: the sum: 3 -8 = | the scalar multiple: -2 3 = | the zero vector: = | the additive inverse of x|: x = | Solution As per the definition of binary operation 3+ (-8) = 3-8+3 = -2 ------------------------ -2.3 = -2(3)+3(-2)-3 = -15 -------------------------- u+v = u+v+3 = v if u = -3 Hence zero vector = -3 ------------------------------ Inverse of x will be y such that y+x = x+y = x+y+3 =-3 Or y = -6-x Hence inverse of x is -6-x.