Normally an orthogonal basis of a finite vector space is referred as a basis that contains many vectors, i.e. 2 or more.

Consider a vector space that its dimension is 1 - does it have an orthogonal basis?
Is it true to refer to all the bases of that vector space as "orthogonal"?

I didn't find a reference for that in Wikipedia.

  • $\begingroup$ Indeed, that WP article on orthogonal basis is very stubby and does not discuss existence at all. However, the Gram-Schmidt process works in any finite dimensional inner product space, including dimension 1 (and even 0) $\endgroup$ – Hagen von Eitzen Feb 6 '15 at 19:23

You are correct. Any basis for a one dimensional inner product space is an orthogonal basis because the orthogonality condition is vacuously true, i.e. there are no pairs which must be orthogonal.

  • 2
    $\begingroup$ The same applies to zero-dimensional space, "even more vacuously". $\endgroup$ – Hagen von Eitzen Feb 6 '15 at 19:19

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