definition 5.101 External direct sum

open in the book · parts/02-mathematical-methods/03-linear-algebra-representations.tex:4418 · p. 148

ground object -- no derivation owed

Rests on

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 4.3 Algebraic structure ¶
      - depends_on definition 4.2 Binary operation ¶
    - depends_on definition 4.2 Binary operation ¶ ↺
- depends_on definition 4.32 Field ¶
  - depends_on definition 4.11 Inverse element ¶
    - depends_on definition 4.9 Neutral element ¶
      - depends_on definition 4.3 Algebraic structure ¶ ↺
  - depends_on definition 4.20 Monoid ¶
    - depends_on definition 4.9 Neutral element ¶ ↺
    - depends_on definition 4.18 Semigroup ¶
      - depends_on definition 4.6 Associativity ¶
      - depends_on definition 4.4 Internal binary operation; magma ¶ ↺
  - depends_on definition 4.30 Ring ¶
    - depends_on definition 4.8 Commutativity; abelian structure ¶ ↺
    - depends_on definition 4.28 Distributivity ¶
      - depends_on definition 4.3 Algebraic structure ¶ ↺
    - depends_on definition 4.21 Group ¶
      - depends_on definition 4.11 Inverse element ¶ ↺
      - depends_on definition 4.20 Monoid ¶ ↺
    - depends_on definition 4.18 Semigroup ¶ ↺
- depends_on definition 4.31 Module ¶
  - depends_on definition 4.29 Action ¶
    - depends_on definition 4.9 Neutral element ¶ ↺
  - 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 ¶ ↺

Supports

depends_on definition 12.84 External direct sum ¶
- depends_on proposition 12.85 The direct sum is a Hilbert space ¶
  - depends_on proposition 12.87 Expansion in an orthogonal decomposition ¶
    - depends_on lemma A.250 Decomposition into cyclic subspaces ¶
      - depends_on proposition A.280 Direct-integral form of the spectral theorem ¶
depends_on definition 5.103 Sum of subspaces; internal direct sum ¶
- depends_on corollary 5.108 The sum is the right operation ¶
  - depends_on example 5.109 Two lines in physical space ¶
- depends_on proposition 5.104 Criterion for a direct sum ¶
- depends_on proposition 5.105 Grassmann's formula ¶
depends_on proposition 5.102 Dimensions add ¶
- depends_on proposition 5.104 Criterion for a direct sum ¶ ↺

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`	→	Vector space	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:4427
`depends_on`	→	Linear combination	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:4427
`depends_on`	←	External direct sum	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2360
`depends_on`	←	Sum of subspaces; internal direct sum	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:4478
`depends_on`	←	Dimensions add	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:4442