lemma 5.38 Exchange and completion

open in the book · parts/02-mathematical-methods/03-linear-algebra-representations.tex:1743 · p. 117

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 5.12 Subspace generated by a set of vectors ¶ ↺
depends_on definition 5.14 Linear independence ¶ ↺
proves proof ch:03-linear-algebra-representations@proof-11 ¶

Supports

depends_on proposition 5.61 Annihilator of a subspace ¶
- depends_on corollary 5.62 A functional that annihilates a set of constraints ¶
depends_on proposition 5.102 Dimensions add ¶
- depends_on proposition 5.104 Criterion for a direct sum ¶
depends_on proposition 5.74 prop:lin-geometric-le-algebraic ¶
depends_on proposition 5.105 Grassmann's formula ¶
depends_on theorem 5.40 Rank–nullity ¶
- depends_on corollary 5.62 A functional that annihilates a set of constraints ¶ ↺
- depends_on lemma A.610 Block positivity ¶
  - depends_on proposition A.611 The constraint on $\Gamma$ ¶
    - depends_on proposition A.612 The dictionary between $A$ and $\Gamma$ ¶
- depends_on lemma A.600 The real part of an inverse ¶
  - depends_on lemma A.601 Several variables, complex symmetric matrix ¶
    - depends_on proposition A.612 The dictionary between $A$ and $\Gamma$ ¶ ↺
    - depends_on proposition A.602 The Wigner function of a Gaussian is a positive Gaussian ¶
      - depends_on proposition A.604 Non-negativity forbids a vanishing overlap ¶
  - depends_on proposition A.612 The dictionary between $A$ and $\Gamma$ ¶ ↺
- depends_on lemma A.564 Dimension and double orthogonal ¶
  - depends_on proposition A.565 Freeness makes every value regular ¶
    - depends_on lemma A.567 The differential of the momentum map along the orbit ¶
      - depends_on theorem A.568 The radical is the isotropy orbit ¶
    - depends_on proposition A.566 The isotropy group acts, and the quotient is smooth ¶
      - depends_on proposition A.570 Existence and uniqueness of the reduced form ¶
  - depends_on theorem A.568 The radical is the isotropy orbit ¶ ↺
- depends_on lemma 5.97 Fitting splitting ¶
  - depends_on theorem 5.98 Primary decomposition ¶
    - depends_on corollary 5.99 Semisimple and nilpotent parts ¶
- depends_on proposition 5.49 Injective, surjective, invertible ¶
  - depends_on lemma 5.97 Fitting splitting ¶ ↺
  - depends_on proposition 5.31 Gram criterion ¶
  - depends_on proposition 5.2 Jacobi's formula, column form ¶
    - depends_on lemma 44.10 Divergence theorem on $(M,g)$ ¶
      - depends_on proposition 44.11 The boundary term, and the Gibbons–Hawking–York action ¶
    - depends_on lemma 44.7 Variation of inverse and determinant ¶
      - depends_on proposition 44.17 Stress–energy of the electromagnetic field ¶
      - depends_on theorem 44.9 Variation of the Einstein–Hilbert action ¶
    - depends_on proposition 7.111 Jacobi's formula, cofactor form ¶
  - depends_on proposition 5.90 Polar decomposition ¶
  - depends_on proposition 5.130 An orthogonal transformation is an isometry ¶
    - depends_on theorem 5.133 Isotropic Cartesian tensors of rank at most four ¶
      - depends_on lemma A.742 Isotropic representation ¶
      - depends_on lemma A.671 The three products are independent ¶
      - depends_on lemma 30.26 Isotropic Cartesian tensors of rank four ¶
      - depends_on remark A.677 The lower ranks, and why an isotropic solid is not piezoelectric ¶
      - depends_on remark 30.27 Two results this chapter borrows from Part II ¶
      - depends_on remark 13.7 What the classification is used for ¶
  - depends_on theorem 5.71 The eigenvalues are the roots of the characteristic polynomial ¶
    - depends_on corollary 5.72 Existence of an eigenvalue over $\C$ ¶
      - depends_on theorem 5.98 Primary decomposition ¶ ↺
      - depends_on theorem 5.79 Spectral theorem for a self-adjoint operator ¶
    - depends_on proposition 32.17 Classification of planar fixed points ¶
      - depends_on example 32.18 A linear centre that is really a stable focus ¶
      - depends_on theorem 32.25 Hopf bifurcation, quoted ¶
    - depends_on theorem 5.84 Spectral theorem for a real symmetric operator ¶
      - depends_on corollary 5.91 The operator norm of a real array ¶
      - depends_on lemma A.610 Block positivity ¶ ↺
      - depends_on lemma A.601 Several variables, complex symmetric matrix ¶ ↺
      - depends_on lemma A.600 The real part of an inverse ¶ ↺
      - depends_on proposition 5.90 Polar decomposition ¶ ↺
      - depends_on proposition 5.94 Principal axes of a real quadratic form ¶
      - depends_on proposition 5.92 The eigenvalues do not control the norm ¶
      - depends_on theorem 5.86 Simultaneous diagonalization of a definite pencil ¶
    - depends_on theorem 9.26 Structure of the solutions ¶
      - depends_on corollary 9.27 Linear stability ¶
- depends_on theorem 5.43 The four fundamental subspaces ¶
- depends_on theorem 5.79 Spectral theorem for a self-adjoint operator ¶ ↺

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`	→	Basis	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1750
`depends_on`	→	Subspace generated by a set of vectors	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1750
`depends_on`	→	Linear independence	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1750
`depends_on`	←	Annihilator of a subspace	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:2703
`depends_on`	←	Dimensions add	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:4442
`depends_on`	←	prop:lin-geometric-le-algebraic	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:3280
`depends_on`	←	Grassmann's formula	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:4531
`depends_on`	←	Rank–nullity	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1829
`proves`	←	ch:03-linear-algebra-representations@proof-11	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1753