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Thread: Proof question - Orthogonal Vectors 2

  1. #1
    Ife
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    Proof question - Orthogonal Vectors 2

    How do I go about showing this:

    Let $\displaystyle W= Span$ $\displaystyle \left\{ v_1, ....., v_p \right\} $. Show that if $\displaystyle x $ is orthogonal to each $\displaystyle v_j $, for $\displaystyle 1 \leq j \leq p $, then $\displaystyle x $ is orthogonal to every vector in $\displaystyle W $.
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    MHF Contributor FernandoRevilla's Avatar
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    If $\displaystyle v\in \textrm{Span}\{v_1,\ldots,v_p \}$ , there are scalars $\displaystyle \lambda_1,\ldots,\lambda_p$ such that:

    $\displaystyle v=\lambda_1v_1+\ldots+\lambda_pv_p$

    Then,

    $\displaystyle <x,v>=<x,\lambda_1v_1+\ldots+\lambda_pv_p>=\ldots= 0$
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  3. #3
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    you test for orthogonality by taking the _____? product. what's the magic number?

    is this ____? product bi-linear (linear in each variable? or conjugate-linear if you are using complex numbers)?

    if it is, can you break a ____? product down into sums of _____? products?

    is 0+0+...+0 still zero?
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