Given a basis $(v_1,\ldots,v_m)$ for some vector space $V\subseteq\mathbb Q^n$ with $m\le n$, can we always find an orthonormal base for $V$ with rational basis vectors? If yes, how can we construct such a basis?
2026-03-30 17:02:08.1774890128
Can we always find an orthonormal base with rational vectors?
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No we can't. Let $V\subset \mathbb Q^2$ be the line $x=y$. Every vector $(x,x)$ in $V$ has irrational length $\sqrt2 |x|$, so $V$ contains no unit vectors.