Loogle!

Result

Found 12 declarations mentioning Matrix and LinearIndependent.

• Matrix.vecMul_injective_iff 📋 Mathlib.LinearAlgebra.Matrix.ToLin
  {m : Type u_3} {n : Type u_4} [Fintype m] {R : Type u_5} [Ring R] {M : Matrix m n R} : (Function.Injective fun v => Matrix.vecMul v M) ↔ LinearIndependent R M.row
• Matrix.linearIndependent_rows_of_isUnit 📋 Mathlib.LinearAlgebra.Matrix.ToLin
  {m : Type u_3} [Fintype m] {R : Type u_5} [Ring R] {A : Matrix m m R} [DecidableEq m] (ha : IsUnit A) : LinearIndependent R A.row
• Matrix.mulVec_injective_iff 📋 Mathlib.LinearAlgebra.Matrix.ToLin
  {m : Type u_4} {n : Type u_5} [Fintype n] {R : Type u_6} [CommRing R] {M : Matrix m n R} : Function.Injective M.mulVec ↔ LinearIndependent R M.col
• Matrix.linearIndependent_cols_of_isUnit 📋 Mathlib.LinearAlgebra.Matrix.ToLin
  {m : Type u_4} {R : Type u_6} [CommRing R] [Fintype m] {A : Matrix m m R} [DecidableEq m] (ha : IsUnit A) : LinearIndependent R A.col
• Matrix.linearIndependent_cols_of_det_ne_zero 📋 Mathlib.LinearAlgebra.Matrix.Determinant.Basic
  {m : Type u_1} [DecidableEq m] [Fintype m] {R : Type v} [CommRing R] [IsDomain R] {A : Matrix m m R} (hA : A.det ≠ 0) : LinearIndependent R A.col
• Matrix.linearIndependent_rows_of_det_ne_zero 📋 Mathlib.LinearAlgebra.Matrix.Determinant.Basic
  {m : Type u_1} [DecidableEq m] [Fintype m] {R : Type v} [CommRing R] [IsDomain R] {A : Matrix m m R} (hA : A.det ≠ 0) : LinearIndependent R A.row
• Matrix.linearIndependent_cols_iff_isUnit 📋 Mathlib.LinearAlgebra.Matrix.NonsingularInverse
  {m : Type u} [DecidableEq m] {K : Type u_3} [Field K] [Fintype m] {A : Matrix m m K} : LinearIndependent K A.col ↔ IsUnit A
• Matrix.linearIndependent_rows_iff_isUnit 📋 Mathlib.LinearAlgebra.Matrix.NonsingularInverse
  {m : Type u} [DecidableEq m] {K : Type u_3} [Field K] [Fintype m] {A : Matrix m m K} : LinearIndependent K A.row ↔ IsUnit A
• Matrix.linearIndependent_cols_of_invertible 📋 Mathlib.LinearAlgebra.Matrix.NonsingularInverse
  {m : Type u} [DecidableEq m] {K : Type u_3} [Field K] [Fintype m] (A : Matrix m m K) [Invertible A] : LinearIndependent K A.col
• Matrix.linearIndependent_rows_of_invertible 📋 Mathlib.LinearAlgebra.Matrix.NonsingularInverse
  {m : Type u} [DecidableEq m] {K : Type u_3} [Field K] [Fintype m] (A : Matrix m m K) [Invertible A] : LinearIndependent K A.row
• LinearIndependent.rank_matrix 📋 Mathlib.Data.Matrix.Rank
  {m : Type um} {n : Type un} {R : Type uR} [Fintype n] [Field R] [Fintype m] {M : Matrix m n R} (h : LinearIndependent R M.row) : M.rank = Fintype.card m
• SimpleGraph.linearIndependent_lapMatrix_ker_basis_aux 📋 Mathlib.Combinatorics.SimpleGraph.LapMatrix
  {V : Type u_1} [Fintype V] (G : SimpleGraph V) [DecidableRel G.Adj] [DecidableEq V] [DecidableEq G.ConnectedComponent] : LinearIndependent ℝ G.lapMatrix_ker_basis_aux

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using (by default) Ctrl-K Ctrl-S. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

1. By constant:
   🔍 Real.sin
   finds all lemmas whose statement somehow mentions the sine function.

2. By lemma name substring:
   🔍 "differ"
   finds all lemmas that have "differ" somewhere in their lemma name.

3. By subexpression:
   🔍 _ * (_ ^ _)
   finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

   The pattern can also be non-linear, as in
   🔍 Real.sqrt ?a * Real.sqrt ?a

   If the pattern has parameters, they are matched in any order. Both of these will find List.map:
   🔍 (?a -> ?b) -> List ?a -> List ?b
   🔍 List ?a -> (?a -> ?b) -> List ?b

4. By main conclusion:
   🔍 |- tsum _ = _ * tsum _
   finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

   As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
   🔍 |- _ < _ → tsum _ < tsum _
   will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
would find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

The #lucky button will directly send you to the documentation of the first hit.

Source code

You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO.

This is Loogle revision 19971e9 serving mathlib revision f167e8d