Rank & Nullity of a Matrix

The problem, from first principles

05-matrix-inverse established that some matrices simply have no inverse. Rank is the precise number that tells you exactly why, and exactly how badly: it counts how many of a matrix's rows (or, equivalently, columns) are genuinely independent, rather than being a combination of the others. A square matrix has an inverse exactly when its rank equals its size; anything less, and the matrix is throwing away information that can never be recovered, which is exactly what a missing inverse means.

From theory to code

Implement rank(A), computing the number of independent rows of A, then nullity(A), computing A's number of columns minus its rank — the size of the space of vectors A maps to zero. The signature and docstring are already in the editor.

Constraints

• A is an m × n NumPy array (not necessarily square).
• Compute rank via row-reduction (10-gaussian-elimination's approach, extended to handle non-square, rank-deficient inputs by skipping a column entirely when every remaining candidate pivot in it is zero), not by calling np.linalg.matrix_rank directly — the point is to see rank fall directly out of row-reduction, not to treat it as a library black box.
• Use a small numerical tolerance (e.g. abs(value) < 1e-10 counts as zero) rather than exact equality, since floating-point row-reduction rarely produces an exact 0.0.

Hints