Theorems · Theorem · linear algebra

Matrix.Pivot.listTransvecCol_mul_last_col

∀ {𝕜 : Type u_3} [inst : Field 𝕜] {r : ℕ} (M : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜),
  M (Sum.inr ()) (Sum.inr ()) ≠ 0 →
    ∀ (i : Fin r), ((Matrix.Pivot.listTransvecCol M).prod * M) (Sum.inl i) (Sum.inr ()) = 0

Multiplying by all the matrices in listTransvecCol M kills all the coefficients in the last column but the last one.

Defined in: Mathlib.LinearAlgebra.Matrix.Transvection
Cited by: 1 results in Mathlib
Foundations: Depth 80 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: Field

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Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Fieldstatement and proof · cited by 7,404
Matrixstatement and proof · cited by 4,303
mul_oneproof · cited by 3,885
LE.le.transproof · cited by 3,151
one_mulproof · cited by 2,841
MulZeroClass.mul_zeroproof · cited by 2,091
one_ne_zeroproof · cited by 885
neg_mulproof · cited by 654
div_oneproof · cited by 629
mul_negproof · cited by 590
le_or_gtproof · cited by 269
add_neg_cancelproof · cited by 213

Cited by1

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Matrix.Pivot.listTransvecCol_mul_mul_listTransvecRow_last_rowproof · cited by 1