Linear dependence and inner product space

[Defennder]

Homework Statement

The following is from the book Linear Algebra 3rd Edn by Stephen Friedberg, et al:

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 $$v_1, \ ... \,v_k \in S$$ and $$\sum_{i=1}^k a_i v_i = 0$$

By theorem 6.3, a_j = $$\langle 0,v_j \rangle / ||v_j||^2 = 0$$ for all j. So S is linearly independent.

Here a_j are scalars of field F and v_j are vectors of inner product space V.

Homework Equations

Theorem 6.3:

Let V be an inner product space, and let S = {v₁, ... , v_k} be an orthogonal set of non-zero vectors. If $$y = \sum^k_{i=1} a_i v_i$$ then $$a_j = \langle y, v_j \rangle / ||v_j||^2$$ for all j

The Attempt at a Solution

Now I don't understand why theorem 6.3 implies a_j = 0 = $$\langle 0,v_j \rangle / ||v_j||^2$$ for all j. I can see how this is zero if the numerator $$\langle 0,v_j \rangle$$ = 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?

 

[CompuChip]

[...] the inner product isn't even defined yet

I find that hard to believe, since I don't see how you can talk about an inner product space without having defined an inner product :)
So I guess you should go back a bit and find that definition and you will see that it immediately follows from the properties of an inner product that <v, 0> = 0 for all v. For example:

• Let x be any vector, then <v, 0> = <v, 0*x> (see definition of vector space) = 0 * <v, x> (by linearity of the inner product) = 0 (by multiplicative properties of the reals )
• <v, 0> = <v, v - v> (by definition of the vector space) = <v, v> + <v, -v> (by additivity of the inner product) = <v, v> - <v, v> = 0. Of course, technically, v - v = v + (-v) with the existence of -v asserted by the definition of vector space, etc.

 

[Defennder]

I find that hard to believe, since I don't see how you can talk about an inner product space without having defined an inner product :)

Well I meant the specific inner product function, not the general notion of an inner product on a vector space.

• Let x be any vector, then <v, 0> = <v, 0*x> (see definition of vector space) = 0 * <v, x> (by linearity of the inner product) = 0 (by multiplicative properties of the reals )

This is just what I need. Thanks!