This is probably a basic question, but my linear algebra is weak.

Suppose I want to compute the nullspace of a matrix A using some iterative method (e.g. Lanczos). Suppose further that I know a priori the nullspace of the first n columns of the matrix, i.e., Av = [0 0 0 ... 0 b_n .. B_N], where b_i are nonzero with high probability.

Does starting the iterative method with vector v (instead of a random vector) speed the iterative method (e.g., Lanczos) up at all?