1. The problem statement, all variables and given/known data S = a non-empty set of vecotrs in V S' = set of all vectors in V which are orthogonal to every vector in S Show S' = subspace of V 2. Relevant equations Subspace requirements. 1. 0 vector is there 2. Closure under addition 3. Closure under scalar multiplication 3. The attempt at a solution 1. 0 is held within S' by definition here 2. Let v and v' exist within S' and let s exist within S. <v+v',s>=<v,s>+<v',s> = 0+0 =0 which is what we want since <v,s>=0 and <v',s>=0. 3. if x is a scalar, then <xv,s> = x<v,s> =x*0 = 0 therefore we have satisfied all the necessary requirements and have shown that S' is a subspace of V. Question: In my notes, the instructor lists a fourth requirement, namely If v exists within S, S a subspace, then -v exists within s. Must I prove this as well?