proposition 5.49 Injective, surjective, invertible

open in the book · parts/02-mathematical-methods/03-linear-algebra-representations.tex:2295 · p. 124

Rests on

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 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 5.7 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 ¶

Supports

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 5.31 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 proposition 44.38 The electromagnetic field satisfies NEC, WEC, SEC and DEC ¶
  - depends_on theorem 44.9 Variation of the Einstein–Hilbert action ¶
    - depends_on proposition 44.11 The boundary term, and the Gibbons–Hawking–York action ¶ ↺
    - depends_on proposition 44.30 The cosmological term ¶
      - depends_on proposition 44.31 de Sitter geometry ¶
- 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 corollary A.743 Two scalar functions instead of a tensor field ¶
      - depends_on lemma A.748 The second-order structure function ¶
      - depends_on lemma A.747 The third-order structure functions ¶
      - 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 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 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 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 ¶
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 definition 30.20 Pressure and deviatoric stress ¶
      - depends_on definition 30.31 The engineering moduli ¶
      - depends_on definition 30.79 Maxwell viscoelasticity ¶
      - depends_on proposition 30.74 The maximum shear stress, and its indifference to pressure ¶
      - depends_on remark 30.75 Two criteria, one uncited ¶
    - depends_on proposition 30.9 Principal strains ¶
      - depends_on example 30.10 Simple shear and pure shear are the same strain ¶
    - depends_on theorem 5.80 Simultaneous diagonalization of commuting self-adjoint operators ¶
      - depends_on corollary 5.82 Spectral theorem for a normal operator ¶
    - 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 28.33 Normal modes ¶
      - depends_on example 28.39 The linear triatomic molecule ¶
      - depends_on proposition 28.40 Modes and dispersion of the loaded string ¶
      - depends_on remark 28.34 Part II owes the simultaneous diagonalization of two quadratic forms ¶
    - depends_on theorem 29.16 Principal axes ¶
      - depends_on definition 29.19 Inertia ellipsoid ¶
      - depends_on definition 29.35 The Lagrange top ¶
      - depends_on definition 29.18 Classification of tops ¶
      - depends_on phenomenon 29.31 The intermediate-axis instability ¶
      - depends_on proposition 29.33 Dissipation drives a free body to its largest moment ¶
      - depends_on proposition 29.17 Triangle inequalities ¶
      - depends_on theorem 29.25 Euler's equations ¶
- 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 proposition 32.26 The Hopf normal form ¶
- depends_on theorem 5.84 Spectral theorem for a real symmetric operator ¶ ↺
- depends_on theorem 9.26 Structure of the solutions ¶
  - depends_on corollary 9.27 Linear stability ¶
    - depends_on corollary 9.41 Hill's equation ¶
      - depends_on proposition 9.43 The unmodulated line, and the feet of the tongues ¶
    - depends_on proposition 9.31 Lyapunov's first method: asymptotic stability ¶
    - depends_on proposition 9.29 The planar classification ¶
    - depends_on theorem 9.39 Floquet ¶
      - depends_on corollary 9.41 Hill's equation ¶ ↺

Neighborhood

Every logical edge within two steps of this node.

declared and complete
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a check failed
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Edges

type	direction	node	provenance	where
`depends_on`	→	Kernel, image, nullity, rank	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:2303
`depends_on`	→	Existence, uniqueness, and linearity of the inverse	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:2303
`depends_on`	→	Rank–nullity	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:2303
`depends_on`	←	Fitting splitting	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:4261
`depends_on`	←	Gram criterion	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1424
`depends_on`	←	Jacobi's formula, column form	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:430
`depends_on`	←	Polar decomposition	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:3958
`depends_on`	←	An orthogonal transformation is an isometry	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:5658
`depends_on`	←	The eigenvalues are the roots of the characteristic polynomial	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:3217
`proves`	←	ch:03-linear-algebra-representations@proof-18	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:2306