1. The problem statement, all variables and given/known data The following is from the book Linear Algebra 3rd Edn by Stephen Friedberg, et al: Here aj are scalars of field F and vj are vectors of inner product space V. 2. Relevant equations Theorem 6.3: 3. The attempt at a solution Now I don't understand why theorem 6.3 implies aj = 0 = [tex]\langle 0,v_j \rangle / ||v_j||^2 [/tex] for all j. I can see how this is zero if the numerator [tex]\langle 0,v_j \rangle[/tex] = 0, but the inner product isn't even defined yet and I don't see anywhere in the axioms that the inner product of the zero vector and an orthogonal vector would always be zero. So how does theorem 6.3 apply here?