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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?

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Yes.

You will need to apply $A$ fewer times to get the linear dependence of the rightmost columns.

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