To confirm my analogy, I am asking if I can consider every orthonormal basis to be an orthogonal one (but not vv) in the same sense that every square is a rectangle (but not vv).

I believe the answer to be yes bc the only difference between an orthonormal basis & its orthogonal variation is that the former's columns are normalized. So in that sense, if someone were to ask a q of the following form...

Find an orthogonal basis for the column space of $A = \left[\begin{array}{ccc} 1 & 0 & 0 \\ 1 & 4 & 0 \\ 1 & 4 & 6 \\ 1 & 4 & 6 \end{array}\right]$.

Then, I (technically) would be correct in providing an orthonormal basis instead?

Forgive me, I know the q is a silly one bc the answer is made obvious in the context of so many resources, but I seek a direct response.


Just so.

As an aside, your analogy is particularly apt in that a rectangle essentially defines an orthogonal basis for $\mathbb R^2$, while a square defines an orthonormal one.


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