So, given a basis for $V$, I know how to use the Gram-Schmidt process to get an orthonormal basis.
My question is, suppose we have an orthonormal basis for a subspace $U$ of $V$. How can we extend this to one for $V$?
Edit: Let $u_1,…,u_n$ be the orthonormal basis for $U$. Usually in Gram Schmidt we start by producing a new set of vectors that maintain orthogonality and spanning. So set the first $n$ vectors of this new set to be those basis vectors of $U$. Then iterate Gram from here? But I don’t know if this is correct or if I am not seeing the correct way?
Best Answer
Complete your orthonormal basis $(u_1,\dots,u_n)$ of $U$ into an ordinary basis $(u_1,\dots,u_n,u_{n+1},\dots,u_m)$ of $V,$ to which you apply Gram-Schmidt. The orthonormal basis of $V$ you obtain will have its $n$ first vectors equal to $u_1,\dots,u_n.$