proposition 5.130 An orthogonal transformation is an isometry

open in the book · parts/02-mathematical-methods/03-linear-algebra-representations.tex:5647 · p. 163

Rests on

depends_on definition 5.18 Inner product ¶
- 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 4.32 Field ¶
    - depends_on definition 4.11 Inverse element ¶
      - depends_on definition 4.9 Neutral element ¶
    - depends_on definition 4.20 Monoid ¶
      - depends_on definition 4.9 Neutral element ¶ ↺
      - depends_on definition 4.18 Semigroup ¶
    - depends_on definition 4.30 Ring ¶
      - depends_on definition 4.8 Commutativity; abelian structure ¶ ↺
      - depends_on definition 4.28 Distributivity ¶
      - depends_on definition 4.21 Group ¶
      - 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 equation 5.44 eq:lin-norm-assoc ¶
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 ¶
      - depends_on definition 4.33 Vector space ¶ ↺
  - 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 5.5 Linear combination ¶ ↺
  - 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-56 ¶

Supports

depends_on theorem 5.133 Isotropic Cartesian tensors of rank at most four ¶
- depends_on lemma A.742 Isotropic representation ¶
  - depends_on corollary A.743 Two scalar functions instead of a tensor field ¶
    - depends_on lemma A.748 The second-order structure function ¶
      - depends_on theorem A.749 Kolmogorov's equation and the four-fifths law ¶
    - depends_on lemma A.747 The third-order structure functions ¶
      - depends_on theorem A.749 Kolmogorov's equation and the four-fifths law ¶ ↺
    - depends_on proposition A.746 The Kármán–Howarth equation in classical form ¶
  - depends_on lemma A.747 The third-order structure functions ¶ ↺
  - depends_on lemma A.744 Vanishing of the pressure–velocity correlation ¶
    - depends_on theorem A.745 Kármán–Howarth–Monin relation ¶
      - depends_on proposition A.746 The Kármán–Howarth equation in classical form ¶ ↺
      - depends_on theorem A.749 Kolmogorov's equation and the four-fifths law ¶ ↺
- depends_on lemma A.671 The three products are independent ¶
  - depends_on proposition A.673 Cubic tensors of rank four ¶
    - depends_on theorem A.675 Three constants for a cubic crystal, two for an isotropic solid ¶
      - depends_on remark A.676 The third cubic constant is measured, and it is not small ¶
- depends_on lemma 30.26 Isotropic Cartesian tensors of rank four ¶
  - depends_on phenomenon 30.28 Hooke's law ¶
    - depends_on phenomenon 30.32 Poisson contraction ¶
      - depends_on proposition 30.33 Conversion among the moduli ¶
    - depends_on remark 30.78 Anharmonicity: what the quadratic energy also suppresses ¶
- 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 ¶

Neighborhood

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partly declared
a check failed
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Edges

type	direction	node	provenance	where
`depends_on`	→	Inner product	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:5658
`depends_on`	→	eq:lin-norm-assoc	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:5658
`depends_on`	→	Injective, surjective, invertible	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:5658
`depends_on`	←	Isotropic Cartesian tensors of rank at most four	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:5750
`proves`	←	ch:03-linear-algebra-representations@proof-56	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:5662