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Homework Help: Linear dependence and inner product space

  1. Aug 2, 2008 #1


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    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?
  2. jcsd
  3. Aug 2, 2008 #2


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    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 :smile:)
    • <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.
  4. Aug 2, 2008 #3


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    Well I meant the specific inner product function, not the general notion of an inner product on a vector space.
    This is just what I need. Thanks!
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