theorem 5.40 Rank–nullity

open in the book · parts/02-mathematical-methods/03-linear-algebra-representations.tex:1823 · p. 118

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 5.39 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 ¶ ↺
- proves proof ch:03-linear-algebra-representations@proof-12 ¶
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 ¶

Supports

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.609 The exponent is a quadratic polynomial ¶
- 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.570 Existence and uniqueness of the reduced form ¶
  - 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 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 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.747 The third-order structure functions ¶
      - depends_on lemma A.744 Vanishing of the pressure–velocity correlation ¶
    - depends_on lemma A.671 The three products are independent ¶
      - depends_on proposition A.673 Cubic tensors of rank four ¶
    - depends_on lemma 30.26 Isotropic Cartesian tensors of rank four ¶
      - depends_on phenomenon 30.28 Hooke's law ¶
    - 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 proposition 30.9 Principal strains ¶
      - depends_on theorem 5.80 Simultaneous diagonalization of commuting self-adjoint operators ¶
      - depends_on theorem 5.84 Spectral theorem for a real symmetric operator ¶
      - depends_on theorem 28.33 Normal modes ¶
      - depends_on theorem 29.16 Principal axes ¶
  - 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 9.31 Lyapunov's first method: asymptotic stability ¶
      - depends_on proposition 9.29 The planar classification ¶
      - depends_on theorem 9.39 Floquet ¶
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:1829
`depends_on`	→	Kernel, image, nullity, rank	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1829
`depends_on`	→	Exchange and completion	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1829
`depends_on`	←	A functional that annihilates a set of constraints	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:2731
`depends_on`	←	Block positivity	declared	appendices/A-long-proofs.tex:29245
`depends_on`	←	The real part of an inverse	declared	appendices/A-long-proofs.tex:28744
`depends_on`	←	Dimension and double orthogonal	declared	appendices/A-long-proofs.tex:27153
`depends_on`	←	Fitting splitting	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:4261
`depends_on`	←	Injective, surjective, invertible	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:2303
`depends_on`	←	The four fundamental subspaces	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1962
`depends_on`	←	Spectral theorem for a self-adjoint operator	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:3432
`proves`	←	ch:03-linear-algebra-representations@proof-13	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1832