Given independent $v_1,\ldots,v_k$, set $u_1=v_1$ and $u_j=v_j-\sum_{i<j}u_i\langle u_i,v_j\rangle/\|u_i\|^2$. Then normalize $q_j=u_j/\|u_j\|$. Each step removes components along earlier directions. Modified Gram–Schmidt or Householder QR is preferable to classical Gram–Schmidt when numerical stability matters.
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