rank_eq_finrank_span_row — Mathlib

English

The rank of A^T A equals the rank of A.

Русский

Ранк A^T A равен рангу A.

LaTeX

$$$ \operatorname{rank}(A^{T} A) = \operatorname{rank}(A). $$$

Lean4

/-- The rank of a matrix is the rank of the space spanned by its rows. -/
theorem rank_eq_finrank_span_row [Field R] [Finite m] (A : Matrix m n R) :
    A.rank = finrank R (Submodule.span R (Set.range A.row)) :=
  by
  cases nonempty_fintype m
  rw [← rank_transpose, rank_eq_finrank_span_cols, col_transpose]

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