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I am trying to solve excercise 8.5 d) from Straumanns book on general relativity (here V is an n-dimensional Minkowski vector space):

Prove that the orthogonal complement of a null vector is an $(n-1)$-dimensional subspace of $V$ in which the inner product is positive semi-definite and has rank $n-2$.

I understand that every null vector is orthogonal to itself or any linearly dependent vector. But can't a spacelike vector also be orthogonal to a null vector? Also how can one show that the inner product of the orthogonal complement to the null vector has rank (n-2)?

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    $\begingroup$ With a suitable choice of the reference frame the null vecotr is always either of the form $(a,0,...,0,a)$ or $(-a,0,...,0,a)$ ($a \neq 0$). Hence the considered space is generated by n-2 spacelike vectors and respectively $(-1,0,...,0,1)$ or $(1,0,...,0,1)$... $\endgroup$
    – Valter Moretti
    Commented Jul 8 at 13:01
  • $\begingroup$ Ah thank you, this cleared things up! $\endgroup$
    – Lenti
    Commented Jul 8 at 13:30

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