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How to construct a vector perpendicular to a bunch of known vectors?

  1. Jul 18, 2014 #1
    Hi,

    Given several vectors, which may be or not be orthogonal to each other, how to construct a vector perpendicular to them? In a sense of inner production being zero.

    To be specific, I have [itex]n[/itex] vectors [itex]v_{N}[/itex] of length [itex]N[/itex], where [itex]n<N[/itex]. So the maximum rank for these vectors is [itex]n[/itex], which leaves space for new vectors perpendicular to all of them. How to construct such a vector? I know Gram-Schmidt process, but it seems it's not what I want.

    Thanks.
     
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  3. Jul 18, 2014 #2

    micromass

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    Why isn't the Gram-Schmidt process what you want?
     
  4. Jul 18, 2014 #3

    WWGD

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  5. Jul 18, 2014 #4
    Thanks. I see. I should use Gram-Schmidt process.
     
  6. Jul 18, 2014 #5

    WWGD

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    You can, but you can also write a matrix M using your vectors, row-reduce, calculate the nullspace of M and then use the fact , by the fundamental theorem of algebrar ,that the nullspace of the matrix is the ortho complement of the row space, and then you can find a basis for the nullspace. Just an alternative.
     
    Last edited: Jul 18, 2014
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