An orthonormal basis in a subspace.

Question

Let an orthonormal basis of $Bbb{R}^n$ be ${e_1,dots,e_n}$, and $U$ be a subspace in $Bbb{R}^n$. Can we construct the orthonormal basis of $U$ by taking randomly from ${e_1,dots,e_n}$?
If possible, would you prove it?

José Carlos Santos · Answer

In general, no, it is not possible. Take $n=2$, let ${e_1,e_2}$ be the standard basis of $Bbb R^2$ and let $U={(x,x)mid xinBbb R}$. Then no subset of ${e_1,e_2}$ is a basis of $U$, since neither $e_1in U$ nor $e_2in U$.

An orthonormal basis in a subspace.

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