TITLE: How to extend an existing orthogonal set of vectors?
QUESTION [6 upvotes]: Suppose I have $k$ vectors in $\mathbb R^n$ that are orthogonal to each other ($k \ll n$). Is there an efficient way to find another vector that is orthogonal to all these given vectors?
If we put the $k$ vectors in a $k \times n$ matrix, denoted by $A$, then the problem is identical to finding a vector in the null space of $A$. Of course one can resort to the method of solving an underdetermined linear system to find the null space of $A$, but by doing this we have got all the vectors that span the null space of $A$. For $k \ll n$, this is definitely not efficient, since I only need one vector in the null space.
I'm wondering if there exists a smarter way.

REPLY [2 votes]: You could normalize the $k$ vectors and make them the columns of a matrix $Q$.  Then $I - QQ^T$ is the projection onto the orthogonal complement of the span of the $k$ vectors.  Any nonzero column of this matrix is orthogonal to the $k$ vectors.