definition 5.39 Kernel, image, nullity, rank

open in the book · parts/02-mathematical-methods/03-linear-algebra-representations.tex:1796 · p. 118

ground object -- no derivation owed

Rests on

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 4.3 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 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 ¶

Supports

depends_on definition 5.69 Eigenvector, eigenvalue, eigenspace ¶
- depends_on corollary 9.25 Normal modes of a diagonalizable system ¶
  - 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 ¶ ↺
  - depends_on proposition 9.29 The planar classification ¶ ↺
- depends_on definition 5.73 Algebraic and geometric multiplicity ¶
  - depends_on proposition 5.74 prop:lin-geometric-le-algebraic ¶
- depends_on definition 9.30 Hyperbolic equilibrium ¶
  - depends_on theorem 9.32 Hartman–Grobman; quoted ¶
    - depends_on theorem 32.15 Hartman–Grobman, restated from Part II ¶
      - depends_on definition 32.22 Bifurcation ¶
      - depends_on example 32.18 A linear centre that is really a stable focus ¶
- depends_on proposition 5.75 Eigenvectors for distinct eigenvalues are independent ¶
  - depends_on theorem 5.77 Criterion for diagonalizability ¶
    - depends_on proposition 32.39 Stability of a periodic orbit ¶
      - depends_on phenomenon 32.87 Universal period doubling ¶
      - depends_on proposition 32.84 Fixed points and the first period doubling ¶
    - depends_on theorem 32.13 Linear stability ¶
      - depends_on proposition 32.65 Fixed points of the Lorenz system and their stability ¶
      - depends_on proposition 32.23 Saddle-node bifurcation ¶
      - depends_on proposition 32.24 Transcritical and pitchfork bifurcations ¶
      - depends_on proposition 28.69 Which branch is stable, and where it ends ¶
- depends_on theorem 5.77 Criterion for diagonalizability ¶ ↺
- 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 corollary 5.99 Semisimple and nilpotent parts ¶
    - 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 lemma 5.97 Fitting splitting ¶
- depends_on theorem 5.98 Primary decomposition ¶ ↺
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 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 proposition 5.49 Injective, surjective, invertible ¶ ↺
- 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`	→	Linear transformation	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1808
`depends_on`	→	Vector subspace	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1808
`depends_on`	←	Eigenvector, eigenvalue, eigenspace	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:3147
`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`	←	Rank–nullity	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1829
`proves`	←	ch:03-linear-algebra-representations@proof-12	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1811