proposition 5.31 Gram criterion

open in the book · parts/02-mathematical-methods/03-linear-algebra-representations.tex:1415 · p. 113

Rests on

depends_on definition 5.14 Linear independence ¶
- 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 4.4 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 5.18 Inner product ¶
- depends_on definition 4.33 Vector space ¶ ↺
depends_on proposition 5.49 Injective, surjective, invertible ¶
- 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 5.7 Vector subspace ¶
    - depends_on definition 5.5 Linear combination ¶ ↺
  - proves proof ch:03-linear-algebra-representations@proof-12 ¶
- 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 5.39 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 ¶
proves proof ch:03-linear-algebra-representations@proof-7 ¶

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type	direction	node	provenance	where
`depends_on`	→	Linear independence	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1424
`depends_on`	→	Inner product	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1424
`depends_on`	→	Injective, surjective, invertible	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1424
`proves`	←	ch:03-linear-algebra-representations@proof-7	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1428