> Can you give an example of any n-dimensional system where you can't make a basis of n perpendicular vectors? Well, yes. In a totally general vector space (in

> Can you give an example of any n-dimensional system where you can't make a basis of n perpendicular vectors? Well, yes. In a totally general vector space (in the mathematical sense of a vector space over a field; it’s the same notion as used in physics but perhaps more general) there’s no such thing as ‘perpendicular’ (aka ‘orthogonal’) — you need an inner product space for that. Of course, (at least in finite dimensions) you can always define such an inner product and you can certainly create orthogonal bases (see: Gram-Schmidt process), but my point was that the notion of dimension doesn’t require it to make sense. Indeed it’s not part of the usual definition; we only need to be able to talk about sets of vectors and their spans.