**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 apg 327 said:Let V be an inner product space, and let S be an orthogonal set of nonzero vectors. Then S is linearly independent. Proof:

Suppose that [tex]v_1, \ ... \,v_k \in S[/tex] and [tex]\sum_{i=1}^k a_i v_i = 0[/tex]

By theorem 6.3, a_{j}= [tex]\langle 0,v_j \rangle / ||v_j||^2 = 0[/tex] for all j. So S is linearly independent.

_{j}are scalars of field F and v

_{j}are vectors of inner product space V.

**2. Relevant equations**

Theorem 6.3:

Let V be an inner product space, and let S = {v_{1}, ... , v_{k}} be an orthogonal set of non-zero vectors. If [tex]y = \sum^k_{i=1} a_i v_i[/tex] then [tex]a_j = \langle y, v_j \rangle / ||v_j||^2[/tex] for all j

**3. The attempt at a solution**

Now I don't understand why theorem 6.3 implies a

_{j}= 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?