lemma 5.97 Fitting splitting

open in the book · parts/02-mathematical-methods/03-linear-algebra-representations.tex:4252 · p. 146

Rests on

depends_on definition 5.39 Kernel, image, nullity, rank ¶
- depends_on definition 5.37 Linear transformation ¶
  - depends_on definition 4.33 Vector space ¶
    - depends_on definition 4.8 Commutativity; abelian structure ¶
      - depends_on definition 4.4 Internal binary operation; magma ¶
    - depends_on definition 4.32 Field ¶
      - depends_on definition 4.11 Inverse element ¶
      - depends_on definition 4.20 Monoid ¶
      - depends_on definition 4.30 Ring ¶
    - depends_on definition 4.31 Module ¶
      - depends_on definition 4.29 Action ¶
      - depends_on definition 4.28 Distributivity ¶
      - depends_on definition 4.21 Group ¶
      - depends_on definition 4.30 Ring ¶ ↺
- depends_on definition 5.7 Vector subspace ¶
  - depends_on definition 5.5 Linear combination ¶
    - depends_on definition 4.33 Vector space ¶ ↺
- proves proof ch:03-linear-algebra-representations@proof-12 ¶
depends_on proposition 5.49 Injective, surjective, invertible ¶
- depends_on definition 5.39 Kernel, image, nullity, rank ¶ ↺
- depends_on proposition 5.48 Existence, uniqueness, and linearity of the inverse ¶
  - depends_on definition 5.47 Inverse of a linear transformation ¶
    - depends_on definition 5.37 Linear transformation ¶ ↺
  - depends_on definition 5.37 Linear transformation ¶ ↺
  - proves proof ch:03-linear-algebra-representations@proof-17 ¶
- depends_on theorem 5.40 Rank–nullity ¶
  - depends_on definition 5.15 Basis ¶
    - depends_on definition 5.12 Subspace generated by a set of vectors ¶
      - depends_on definition 5.5 Linear combination ¶ ↺
      - depends_on definition 5.7 Vector subspace ¶ ↺
      - proves proof ch:03-linear-algebra-representations@proof-3 ¶
    - depends_on definition 5.14 Linear independence ¶
      - depends_on definition 5.5 Linear combination ¶ ↺
  - depends_on definition 5.39 Kernel, image, nullity, rank ¶ ↺
  - depends_on lemma 5.38 Exchange and completion ¶
    - depends_on definition 5.15 Basis ¶ ↺
    - depends_on definition 5.12 Subspace generated by a set of vectors ¶ ↺
    - depends_on definition 5.14 Linear independence ¶ ↺
    - proves proof ch:03-linear-algebra-representations@proof-11 ¶
  - proves proof ch:03-linear-algebra-representations@proof-13 ¶
- proves proof ch:03-linear-algebra-representations@proof-18 ¶
depends_on theorem 5.40 Rank–nullity ¶ ↺
proves proof ch:03-linear-algebra-representations@proof-43 ¶

Supports

depends_on theorem 5.98 Primary decomposition ¶
- depends_on corollary 5.99 Semisimple and nilpotent parts ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Kernel, image, nullity, rank	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:4261
`depends_on`	→	Injective, surjective, invertible	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:4261
`depends_on`	→	Rank–nullity	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:4261
`depends_on`	←	Primary decomposition	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:4308
`proves`	←	ch:03-linear-algebra-representations@proof-43	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:4265