mathlib3 documentation

linear_algebra.free_module.rank

Extra results about module.rank #

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This file contains some extra results not in linear_algebra.dimension.

@[simp]
theorem rank_finsupp (R : Type u) (M : Type v) [ring R] [strong_rank_condition R] [add_comm_group M] [module R M] [module.free R M] (ι : Type w) :
theorem rank_finsupp' (R : Type u) (M : Type v) [ring R] [strong_rank_condition R] [add_comm_group M] [module R M] [module.free R M] (ι : Type v) :
@[simp]
theorem rank_finsupp_self (R : Type u) [ring R] [strong_rank_condition R] (ι : Type w) :

The rank of (ι →₀ R) is (# ι).lift.

theorem rank_finsupp_self' (R : Type u) [ring R] [strong_rank_condition R] {ι : Type u} :

If R and ι lie in the same universe, the rank of (ι →₀ R) is # ι.

@[simp]
theorem rank_direct_sum (R : Type u) [ring R] [strong_rank_condition R] {ι : Type v} (M : ι Type w) [Π (i : ι), add_comm_group (M i)] [Π (i : ι), module R (M i)] [ (i : ι), module.free R (M i)] :
module.rank R (direct_sum ι (λ (i : ι), M i)) = cardinal.sum (λ (i : ι), module.rank R (M i))

The rank of the direct sum is the sum of the ranks.

@[simp]
theorem rank_matrix (R : Type u) [ring R] [strong_rank_condition R] (m : Type v) (n : Type w) [finite m] [finite n] :

If m and n are fintype, the rank of m × n matrices is (# m).lift * (# n).lift.

@[simp]
theorem rank_matrix' (R : Type u) [ring R] [strong_rank_condition R] (m n : Type v) [finite m] [finite n] :

If m and n are fintype that lie in the same universe, the rank of m × n matrices is (# n * # m).lift.

@[simp]
theorem rank_matrix'' (R : Type u) [ring R] [strong_rank_condition R] (m n : Type u) [finite m] [finite n] :

If m and n are fintype that lie in the same universe as R, the rank of m × n matrices is # m * # n.

@[simp]

The rank of M ⊗[R] N is (module.rank R M).lift * (module.rank R N).lift.

If M and N lie in the same universe, the rank of M ⊗[R] N is (module.rank R M) * (module.rank R N).