  The Restricted Invertibility problem is the problem of selecting the largest
subset of columns of a given matrix $X$, while keeping the smallest singular
value of the extracted submatrix above a certain threshold. In this paper, we
address this problem in the simpler case where $X$ is a random matrix but with
the additional constraint that the selected columns be almost orthogonal to a
given vector $v$. Our main result is a lower bound on the number of columns we
can extract from a normalized i.i.d. Gaussian matrix for the worst $v$.
