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count_affine_solutions_explicit

File dumas2017dual/lib/rouche_capelli.v

Section affine_solution_counting

Type Lemma

Proof Lines 6

Signature

Lemma count_affine_solutions_explicit (x0 : 'rV[K]_m) : x0 *m A = b -> #|affine_solutions| = (#|{:K}| ^ (m - \rank A))%N.

Description

Explicit affine solution count: |{x | xA = b}| = |K|^(m - rank(A)). Combines count_affine_solutions with count_kernel_vectors. This is the main Rouché-Capelli counting formula for row systems.

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