Lax Lean Archive

While this submission is a draft, it cannot be used by other submissions.

Mode space span deficits

Lax342547.SpanDeficits · concepts/Lax342547/SpanDeficits.lean · lax-342547

proven

Loading review…

Community review

Flags

Each flag is tied to a public ORCID identity and explains why this concept may be incorrect.

No flags have been submitted.

    Community review

    Flag this concept

    State precisely what appears incorrect. This explanation will be public under your ORCID name.

    No source line selected.

    Natural Language Statement

    Lemma

    The kernel dimension of the addition map is the difference between the sum of component dimensions and the dimension of their span.

    Concept map
    2 concepts
    100%
    Proven claimThis conceptRelated conceptA → B: B builds on ADescendants are omitted for concepts with more than 10 descendants.
    Evidence

    This concept declares 3 statements. Each proof establishes one of them relative to its assumptions.

    1 addition_kernel_deficit proven

    2 addition_range proven

    3 addition_single proven

    Lean source view on GitHub

    1import Lax342547.RestrictionRank
    2import Mathlib.LinearAlgebra.Pi
    3
    4/-!
    5---
    6title: Mode space span deficits
    7type: lemma
    8---
    9The kernel dimension of the addition map is the difference between the sum of component dimensions and the dimension of their span.
    10-/
    11
    12namespace Lax342547.SpanDeficits
    13
    14open scoped BigOperators
    15
    16noncomputable def addition {K V ι : Type} [Field K] [Fintype ι]
    17 [AddCommGroup V] [Module K V] (S : ι → Submodule K V) : (∀ i, S i) →ₗ[K] V := by
    18 classical
    19 exact LinearMap.lsum K (fun i => S i) K (fun i => (S i).subtype)
    20
    21axiom addition_single {K V ι : Type} [Field K] [Fintype ι] [DecidableEq ι]
    22 [AddCommGroup V] [Module K V] (S : ι → Submodule K V) (i : ι) (v : S i) :
    23 addition S (Pi.single i v) = v.val
    24
    25axiom addition_range {K V ι : Type} [Field K] [Fintype ι]
    26 [AddCommGroup V] [Module K V] (S : ι → Submodule K V) :
    27 LinearMap.range (addition S) = ⨆ i, S i
    28
    29axiom addition_kernel_deficit {K V ι : Type} [Field K] [Fintype ι]
    30 [AddCommGroup V] [Module K V] [FiniteDimensional K V] (S : ι → Submodule K V) :
    31 Module.finrank K (LinearMap.ker (addition S)) =
    32 (∑ i, Module.finrank K (S i))-Module.finrank K (⨆ i, S i : Submodule K V)
    33
    34end Lax342547.SpanDeficits
    35
    Show ProofShow ProofShow Proof
    Builds on
    Used by
    From Mathlib

    Discussion

    Ask a question or add context. Endorsements and structured flags are kept in the review panel above.

    Loading discussion…