proposition 5.102 Dimensions add

open in the book · parts/02-mathematical-methods/03-linear-algebra-representations.tex:4435 · p. 148

Rests on

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 4.33 Vector space ¶
      - depends_on definition 4.8 Commutativity; abelian structure ¶
      - depends_on definition 4.32 Field ¶
      - depends_on definition 4.31 Module ¶
  - depends_on definition 5.7 Vector subspace ¶
    - depends_on definition 5.5 Linear combination ¶ ↺
  - 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 definition 4.33 Vector space ¶ ↺
- depends_on definition 5.5 Linear combination ¶ ↺
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 ¶

Every logical edge within two steps of this node.

type	direction	node	provenance	where
`depends_on`	→	Basis	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:4442
`depends_on`	→	External direct sum	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:4442
`depends_on`	→	Exchange and completion	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:4442
`depends_on`	←	Criterion for a direct sum	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:4496
`proves`	←	ch:03-linear-algebra-representations@proof-46	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:4445