proposition 5.104 Criterion for a direct sum

open in the book · parts/02-mathematical-methods/03-linear-algebra-representations.tex:4481 · p. 149

Rests on

depends_on definition 5.103 Sum of subspaces; internal direct sum ¶
- depends_on definition 5.101 External direct sum ¶
  - 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 5.5 Linear combination ¶
    - depends_on definition 4.33 Vector space ¶ ↺
- depends_on definition 5.7 Vector subspace ¶
  - depends_on definition 5.5 Linear combination ¶ ↺
depends_on definition 5.53 Isomorphism ¶
- depends_on definition 5.37 Linear transformation ¶
  - depends_on definition 4.33 Vector space ¶ ↺
depends_on proposition 5.102 Dimensions add ¶
- 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 5.101 External direct sum ¶ ↺
- 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-46 ¶
proves proof ch:03-linear-algebra-representations@proof-47 ¶

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type	direction	node	provenance	where
`depends_on`	→	Sum of subspaces; internal direct sum	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:4496
`depends_on`	→	Isomorphism	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:4496
`depends_on`	→	Dimensions add	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:4496
`proves`	←	ch:03-linear-algebra-representations@proof-47	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:4500