  Randomized algorithms in numerical linear algebra can be fast, scalable and
robust. This paper examines the effect of sketching on the right singular
vectors corresponding to the smallest singular values of a tall-skinny matrix.
We analyze a fast algorithm by Gilbert, Park and Wakin for finding the trailing
right singular vectors using randomization by examining the quality of the
solution using multiplicative perturbation theory. For an $m\times n$ ($m\geq
n$) matrix, the algorithm runs with complexity $O(mn\log n +n^3)$ which is
faster than the standard $O(mn^2)$ methods. In applications, numerical
experiments show great speedups including a $30\times$ speedup for the AAA
algorithm and $10\times$ speedup for the total least squares problem.
