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Homework Statement
The following is from the book Linear Algebra 3rd Edn by Stephen Friedberg, et al:
pg 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, aj = [tex]\langle 0,v_j \rangle / ||v_j||^2 = 0[/tex] for all j. So S is linearly independent.
Here aj are scalars of field F and vj are vectors of inner product space V.
Homework Equations
Theorem 6.3:
Let V be an inner product space, and let S = {v1, ... , vk} 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
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?