definition 5.15 Basis

open in the book · parts/02-mathematical-methods/03-linear-algebra-representations.tex:833 · p. 106

ground object -- no derivation owed

Rests on

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 4.4 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 ¶ ↺
- proves proof ch:03-linear-algebra-representations@proof-3 ¶
depends_on definition 5.14 Linear independence ¶
- depends_on definition 5.5 Linear combination ¶ ↺

Supports

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 7.97 Partial derivative; gradient ¶
- depends_on definition 7.98 Functions of class $C^{1}$ ¶
  - depends_on corollary 7.113 Inverse function theorem ¶
    - depends_on definition A.503 The substitution property ¶
      - depends_on lemma A.506 Locality ¶
      - depends_on lemma A.504 Transitivity ¶
    - depends_on lemma 7.116 Functions vanishing on a regular zero set ¶
    - depends_on lemma A.510 Every diffeomorphism factorises locally ¶
      - depends_on proposition A.511 The substitution property is universal ¶
    - depends_on remark 7.114 Where the proof is, and why ¶
    - depends_on theorem 7.129 Change of variables in a multiple integral ¶
      - depends_on example 7.130 The two Jacobians this treatise uses ¶
      - depends_on lemma A.518 The excised ball ¶
      - depends_on proposition A.519 The convolution exists ¶
    - depends_on theorem 7.115 Constant rank ¶
  - depends_on definition 7.119 Envelope of a family ¶
    - depends_on proposition 7.120 The envelope touches every member it meets ¶
  - depends_on definition 7.126 Line and surface integrals ¶
    - depends_on corollary A.313 Stokes' theorem for an antisymmetric tensor field ¶
    - depends_on definition 30.17 Traction ¶
      - depends_on proposition 30.19 The stress tensor is symmetric ¶
      - depends_on theorem 30.18 Cauchy: the stress tensor exists ¶
  - depends_on definition 7.127 Simple regions ¶
    - depends_on lemma A.512 The boundary strip is thin ¶
      - depends_on theorem A.513 Change of variables ¶
    - depends_on remark A.516 The hypotheses of the global form ¶
    - depends_on theorem 7.133 Gauss ¶
      - depends_on lemma A.708 The disturbance flux vanishes ¶
      - depends_on lemma A.712 The force is a far-field integral ¶
      - depends_on lemma A.117 Laplacian in orthogonal coordinates ¶
      - depends_on lemma 44.10 Divergence theorem on $(M,g)$ ¶
      - depends_on lemma 31.11 Transport theorem for a material volume ¶
      - depends_on lemma 106.86 A shift of a divergent integral leaves a surface term ¶
      - depends_on lemma 10.44 Green's identities ¶
      - depends_on lemma 10.58 Darboux's equation for spherical means ¶
      - depends_on proposition 23.54 The geometrical amplitude diverges ¶
      - depends_on proposition 10.57 Energy in a backward cone ¶
      - depends_on theorem 16.36 Euler–Lagrange equations for several independent variables ¶
      - depends_on theorem 32.6 Evolution of phase volume ¶
      - … 1 more
    - depends_on theorem 7.131 Green ¶
      - depends_on proposition 32.33 Bendixson's negative criterion ¶
      - depends_on proposition 9.36 Bendixson–Dulac negative criterion ¶
      - depends_on theorem 7.132 Stokes ¶
      - depends_on theorem 8.12 Cauchy ¶
      - depends_on theorem 13.41 Holonomy equals the enclosed curvature; local Gauss–Bonnet ¶
      - depends_on theorem 10.96 Rankine–Hugoniot condition ¶
  - depends_on definition A.508 Primitive map ¶
    - depends_on lemma A.510 Every diffeomorphism factorises locally ¶ ↺
    - depends_on lemma A.509 Primitive maps have the substitution property ¶
      - depends_on proposition A.511 The substitution property is universal ¶ ↺
  - depends_on definition A.116 Orthogonal curvilinear coordinates; scale factors ¶
    - depends_on definition A.125 Ellipsoidal coordinates ¶
      - depends_on lemma A.126 Cartesian coordinates and scale factors ¶
    - depends_on definition A.118 Simple separation of the Helmholtz equation ¶
      - depends_on theorem A.120 Stäckel and Robertson conditions ¶
    - depends_on lemma A.117 Laplacian in orthogonal coordinates ¶ ↺
  - depends_on definition A.298 Half-space; smoothness on it ¶
    - depends_on definition A.299 Manifold with boundary ¶
      - depends_on lemma A.300 The boundary is well defined, and is a manifold ¶
      - depends_on lemma A.307 Partition of unity on a compact manifold ¶
  - depends_on definition 16.11 Admissible class; functional ¶
    - depends_on definition 16.59 Weierstrass excess function ¶
    - depends_on definition 16.12 The two norms; weak and strong neighbourhoods ¶
      - depends_on definition 16.13 Weak and strong extrema ¶
    - depends_on definition 16.15 Variation; the first variation ¶
      - depends_on definition 16.83 One-parameter transformation group; variational symmetry ¶
      - depends_on lemma 16.18 Fundamental lemma ¶
      - depends_on proposition 16.16 Stationarity is necessary ¶
      - depends_on theorem 16.43 Euler's rule for integral constraints ¶
    - depends_on proposition 16.68 Weierstrass' counterexample ¶
  - depends_on definition 10.1 Partial differential equation; order ¶
    - depends_on definition A.102 Normal Cauchy problem of order $k$ ¶
      - depends_on definition A.103 First-order quasilinear system with zero data ¶
      - depends_on lemma A.105 Formal reduction of a normal problem ¶
    - depends_on definition 10.2 Linear, semilinear, quasilinear ¶
      - depends_on definition A.103 First-order quasilinear system with zero data ¶ ↺
      - depends_on definition 10.93 Scalar conservation law ¶
      - depends_on definition 10.3 Principal part and principal symbol ¶
      - depends_on lemma 10.28 The separation constant ¶
      - depends_on proposition 28.44 Superposition ¶
      - depends_on proposition 10.15 Method of characteristics, first order ¶
      - depends_on proposition 10.43 Representation of the solution ¶
      - depends_on theorem 10.22 Cauchy–Kovalevskaya ¶
      - depends_on theorem 10.52 Duhamel's principle ¶
    - depends_on definition 10.24 Well-posed problem ¶
      - depends_on corollary A.152 Uniqueness ¶
      - depends_on corollary 10.76 Uniqueness and stability for the Dirichlet problem ¶
      - depends_on corollary 10.81 Uniqueness for the heat equation ¶
      - depends_on definition 30.22 Boundary conditions of elastostatics ¶
      - depends_on proposition 10.25 The backward heat problem is ill posed ¶
      - depends_on theorem 44.45 Local existence and uniqueness; Choquet-Bruhat; imported ¶
  - depends_on definition 10.66 Harmonic function ¶
    - depends_on theorem A.135 Converse of the mean-value property ¶
      - depends_on corollary A.146 Locally uniform limits of harmonic functions ¶
      - depends_on corollary A.145 Harmonic functions are smooth ¶
    - depends_on theorem A.491 Dirichlet problem for the disc; quoted ¶
    - depends_on theorem 10.69 Mean-value property ¶
      - depends_on corollary A.145 Harmonic functions are smooth ¶ ↺
      - depends_on theorem A.135 Converse of the mean-value property ¶ ↺
      - depends_on theorem 10.77 Strong maximum principle for harmonic functions ¶
    - depends_on theorem 10.75 Weak maximum principle ¶
      - depends_on corollary 10.76 Uniqueness and stability for the Dirichlet problem ¶ ↺
      - depends_on theorem A.525 Helmholtz decomposition: uniqueness ¶
      - depends_on theorem 10.80 Parabolic maximum principle ¶
  - 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 lemma A.520 A $C^{1}$ limit ¶
    - depends_on proposition A.521 Every derivative passes onto $g$ ¶
      - depends_on proposition A.524 Rate of decay ¶
      - depends_on proposition A.522 The Newtonian potential inverts the Laplacian ¶
  - … 8 more
- depends_on definition 7.102 Differential of a function ¶
  - depends_on proposition 7.104 Chain rule in several variables ¶
    - depends_on corollary 7.113 Inverse function theorem ¶ ↺
    - depends_on corollary 7.110 Variable limits of integration ¶
    - depends_on corollary A.292 Inverse function theorem ¶
      - depends_on lemma A.549 The action is transitive ¶
      - depends_on lemma A.300 The boundary is well defined, and is a manifold ¶ ↺
      - depends_on proposition A.535 Existence of a slice ¶
      - depends_on proposition 22.32 Duality of the two brackets ¶
      - depends_on proposition 22.3 Invertibility of the Legendre map ¶
      - depends_on proposition 13.132 Simultaneous straightening of commuting fields ¶
      - depends_on remark 22.19 Which form exists ¶
      - depends_on theorem 13.63 Constant rank theorem ¶
    - depends_on definition 30.4 Material and spatial descriptions ¶
    - depends_on definition 31.4 Material derivative ¶
      - depends_on definition 31.6 Streamline, pathline, streakline ¶
      - depends_on lemma 31.11 Transport theorem for a material volume ¶ ↺
    - depends_on definition 22.26 Lagrange bracket ¶
    - depends_on definition 10.18 Complete integral; envelope ¶
      - depends_on proposition 10.19 An envelope of solutions is a solution ¶
    - depends_on lemma A.510 Every diffeomorphism factorises locally ¶ ↺
    - depends_on lemma A.504 Transitivity ¶ ↺
    - depends_on lemma A.306 The standard smooth bump ¶
      - depends_on lemma A.307 Partition of unity on a compact manifold ¶ ↺
    - depends_on proposition 7.136 Properties of conservative fields ¶
      - depends_on example 31.15 The stream function of a plane flow ¶
      - depends_on proposition 31.28 Potential flow reduces to Laplace's equation ¶
      - depends_on theorem 16.63 Hilbert's invariant integral ¶
      - depends_on theorem 30.15 Compatibility is sufficient on a simply connected body ¶
    - depends_on proposition 7.120 The envelope touches every member it meets ¶ ↺
    - … 9 more
  - depends_on remark 7.103 rem:ana-differential-scalar ¶
- depends_on definition 7.119 Envelope of a family ¶ ↺
- depends_on definition A.70 Pullback ¶
  - depends_on definition 24.17 Symplectomorphism ¶
    - depends_on definition 25.6 Infinitesimal canonical transformation ¶
      - depends_on definition 25.11 Generators of the Galilei transformations ¶
    - depends_on definition 24.28 Ball and cylinder ¶
      - depends_on definition 24.33 Symplectic capacity ¶
      - depends_on example 24.29 A volume-preserving squeeze ¶
      - depends_on proposition 24.30 Linear non-squeezing ¶
      - depends_on theorem 24.31 Gromov's non-squeezing theorem ¶
    - depends_on definition 24.43 Momentum map ¶
      - depends_on definition A.561 Coadjoint action and equivariance ¶
      - depends_on example 24.44 Linear and angular momentum ¶
      - depends_on lemma 24.46 The level set of the momentum map is the symplectic orthogonal of the orbit ¶
      - depends_on theorem 24.45 Noether, symplectic form ¶
    - depends_on theorem 24.18 Symplectic form of the transformation condition ¶
      - depends_on corollary 24.19 Canonical transformations preserve phase volume ¶
  - depends_on lemma A.75 Differentiating a pullback along a flow ¶
    - depends_on lemma A.77 Poincaré lemma, converse form ¶
  - depends_on lemma A.71 The pullback is an algebra map commuting with $\dd$ ¶
- depends_on definition 32.19 Lyapunov function ¶
  - depends_on theorem 32.20 Lyapunov's direct method ¶
    - depends_on example 32.21 The damped pendulum ¶
- depends_on definition 30.2 Representative volume element; continuum field ¶
  - depends_on definition 30.4 Material and spatial descriptions ¶ ↺
  - depends_on definition 30.17 Traction ¶ ↺
  - depends_on example 30.3 How small a piece of copper is still a continuum ¶
  - depends_on remark 30.5 Where the hypothesis fails, and how one can tell ¶
- depends_on definition 22.29 Poisson bracket ¶
  - depends_on definition 26.20 Dirac bracket ¶
    - depends_on definition 26.23 Gauge conditions ¶
      - depends_on proposition 26.24 The Faddeev–Popov determinant ¶
    - depends_on example 26.22 Holonomic constraints ¶
    - depends_on theorem A.617 Jacobi identity for the Dirac bracket ¶
    - depends_on theorem 26.21 Properties of the Dirac bracket ¶
      - depends_on postulate 26.47 Dirac quantization ¶
  - depends_on definition 26.12 First and second class ¶
    - depends_on proposition 26.13 The first-class functions close ¶
      - depends_on proposition 26.43 The hypersurface-deformation algebra ¶
      - depends_on proposition 26.37 Constraint structure of Yang–Mills theory ¶
    - depends_on proposition 26.19 Second-class constraints come in pairs ¶
      - depends_on definition 26.20 Dirac bracket ¶ ↺
      - depends_on proposition 26.24 The Faddeev–Popov determinant ¶ ↺
      - depends_on proposition 26.25 The Liouville measure of a second-class surface ¶
      - depends_on theorem 26.17 Counting the physical degrees of freedom ¶
      - depends_on theorem 26.21 Properties of the Dirac bracket ¶ ↺
    - depends_on proposition 26.34 Proca constraints are second class ¶
      - depends_on proposition 113.3 Degrees of freedom before and after ¶
  - depends_on definition 26.8 Total Hamiltonian ¶
    - depends_on definition 26.16 Extended Hamiltonian ¶
    - depends_on theorem 26.21 Properties of the Dirac bracket ¶ ↺
    - depends_on theorem 26.14 First-class constraints generate gauge transformations ¶
      - depends_on definition 26.16 Extended Hamiltonian ¶ ↺
      - depends_on postulate 26.47 Dirac quantization ¶ ↺
      - depends_on theorem 26.17 Counting the physical degrees of freedom ¶ ↺
  - depends_on definition 25.11 Generators of the Galilei transformations ¶ ↺
  - depends_on definition 25.6 Infinitesimal canonical transformation ¶ ↺
  - depends_on definition 25.2 Poisson algebra ¶
  - depends_on definition 24.36 Poisson manifold ¶
    - depends_on definition 25.2 Poisson algebra ¶ ↺
    - depends_on definition 24.37 Casimir function ¶
      - depends_on proposition 24.41 The free rigid body is a Lie–Poisson system ¶
      - depends_on remark 29.27 Euler's equations as a Lie–Poisson system ¶
      - depends_on theorem 24.38 Symplectic foliation, quoted ¶
    - depends_on definition 24.40 Lie–Poisson bracket ¶
      - depends_on proposition 24.41 The free rigid body is a Lie–Poisson system ¶ ↺
      - depends_on remark 29.27 Euler's equations as a Lie–Poisson system ¶ ↺
    - depends_on theorem 24.38 Symplectic foliation, quoted ¶ ↺
  - depends_on notation 26.28 Field systems ¶
    - depends_on definition 26.40 The $3+1$ decomposition ¶
      - depends_on lemma A.623 The adapted frame of the $3+1$ split ¶
      - depends_on proposition 26.41 The ADM form of the Einstein–Hilbert Lagrangian ¶
    - depends_on notation A.636 Densities and smearing ¶
      - depends_on lemma A.637 Closed form of $H_{\parallel}$, and what it generates ¶
    - depends_on notation 26.36 Gauge fields and their SI dimensions ¶
      - depends_on proposition 26.37 Constraint structure of Yang–Mills theory ¶ ↺
  - depends_on notation 26.1 Weak and strong equality ¶
    - depends_on notation 26.28 Field systems ¶ ↺
  - depends_on proposition 24.15 Poisson bracket from the symplectic form ¶
    - depends_on definition 24.37 Casimir function ¶ ↺
    - depends_on theorem 24.45 Noether, symplectic form ¶ ↺
- depends_on lemma A.73 Differentiation under the integral sign ¶
  - depends_on lemma A.77 Poincaré lemma, converse form ¶ ↺
depends_on definition 5.17 Components of a vector ¶
depends_on definition 5.16 Dimension ¶
- depends_on definition 5.142 Dimension of a representation ¶
- depends_on proposition 5.54 prop:lin-same-dim-isomorphic ¶
depends_on definition 5.60 Functional specified on a basis ¶
depends_on definition 5.26 Orthogonal basis ¶
- depends_on definition 5.27 Orthonormal basis ¶
  - depends_on definition 12.26 Orthonormal system; Fourier coefficients ¶
    - depends_on definition 12.29 Orthonormal basis ¶
      - depends_on definition 12.86 Internal orthogonal decomposition ¶
    - depends_on proposition 12.27 Best approximation and Bessel's inequality ¶
      - depends_on theorem 12.30 Completeness, expansion, Parseval ¶
    - depends_on proposition 12.28 Convergence criterion for orthogonal series ¶
      - depends_on theorem 12.30 Completeness, expansion, Parseval ¶ ↺
  - depends_on definition 5.32 Levi–Civita symbol; cross product ¶
    - depends_on definition 5.131 Isotropic Cartesian tensor ¶
      - depends_on definition A.672 The octahedral rotation group ¶
      - depends_on lemma 5.132 Parity constraint ¶
    - depends_on lemma 5.34 Contraction of two Levi–Civita symbols ¶
      - depends_on proposition 5.35 The identities of the vector algebra of $\R^{3}$ ¶
    - depends_on lemma 5.33 Determinant through the Levi–Civita symbol ¶
      - depends_on proposition 5.35 The identities of the vector algebra of $\R^{3}$ ¶ ↺
      - depends_on theorem 5.133 Isotropic Cartesian tensors of rank at most four ¶
  - depends_on proposition 12.23 Gram–Schmidt in a Hilbert space ¶
    - depends_on corollary 12.24 Separable spaces have countable orthonormal families ¶
      - depends_on theorem 12.33 Every separable Hilbert space is $\ell^{2}$ ¶
    - depends_on example 12.25 Orthogonal polynomials ¶
    - depends_on proposition 12.95 The tensor inner product is well defined and positive definite ¶
      - depends_on example 12.100 Entangled vectors exist ¶
      - depends_on proposition 12.96 Operators on a tensor product ¶
  - depends_on proposition 5.28 Gram–Schmidt ¶
    - depends_on corollary 5.29 Orthogonal decomposition ¶
      - depends_on proposition 5.153 prop:rep-unitary-completely-reducible ¶
      - depends_on theorem 5.43 The four fundamental subspaces ¶
    - depends_on proposition 12.23 Gram–Schmidt in a Hilbert space ¶ ↺
    - depends_on proposition 5.42 The adjoint exists, is unique, and is linear ¶
      - depends_on corollary 5.82 Spectral theorem for a normal operator ¶
    - depends_on proposition 5.155 Averaging trick ¶
      - depends_on theorem 101.62 The mixing matrix is unitary ¶
    - depends_on theorem 5.86 Simultaneous diagonalization of a definite pencil ¶
      - depends_on corollary 5.89 Rayleigh–Ritz ¶
      - depends_on proposition 5.88 Rayleigh quotient of a definite pencil ¶
    - 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 example 5.9 The matrices of a given shape ¶
depends_on lemma 5.38 Exchange and completion ¶
- 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 lemma A.600 The real part of an inverse ¶ ↺
  - 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 proposition A.566 The isotropy group acts, and the quotient is smooth ¶
    - 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 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 lemma 44.7 Variation of inverse and determinant ¶
      - 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 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 proposition 32.17 Classification of planar fixed points ¶
      - depends_on theorem 5.84 Spectral theorem for a real symmetric operator ¶ ↺
      - depends_on theorem 9.26 Structure of the solutions ¶
  - depends_on theorem 5.43 The four fundamental subspaces ¶ ↺
  - 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 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 lemma 5.137 The alternating top form is unique up to scale ¶
- depends_on theorem 5.138 A symplectic transformation has determinant $+1$ ¶
  - depends_on corollary 22.24 Invariance of the phase-space volume ¶
    - depends_on remark 22.25 This is Liouville's theorem ¶
  - depends_on corollary 24.19 Canonical transformations preserve phase volume ¶ ↺
  - depends_on proposition 24.5 Properties of the symplectic group ¶
    - depends_on remark 24.6 Connectedness, and what it rests on ¶
    - depends_on remark 24.7 Contrast with the orthogonal group ¶
depends_on proposition 5.102 Dimensions add ¶ ↺
depends_on proposition 5.105 Grassmann's formula ¶ ↺
depends_on proposition 5.45 prop:lin-matrix-unique ¶
- depends_on proposition 5.125 prop:lin-matrix-algebra ¶
  - depends_on corollary 5.168 cor:rep-adjoint-faithful ¶
  - depends_on definition 5.162 Representation of an algebra ¶
    - depends_on definition 5.163 Faithful, equivalent, invariant ¶
      - depends_on corollary 5.168 cor:rep-adjoint-faithful ¶ ↺
      - depends_on proposition 5.164 The kernel is an ideal ¶
    - depends_on proposition 5.167 Adjoint representation ¶
      - depends_on corollary 5.168 cor:rep-adjoint-faithful ¶ ↺
    - depends_on proposition 5.164 The kernel is an ideal ¶ ↺
  - depends_on definition 9.21 Matrix exponential ¶
    - depends_on proposition 9.22 The exponential and its derivative ¶
      - depends_on theorem 9.23 Solution of a constant-coefficient system ¶
  - depends_on proposition 5.139 Dimension of the symplectic group ¶
    - depends_on proposition 24.5 Properties of the symplectic group ¶ ↺
    - depends_on remark 22.21 The count ¶
- depends_on proposition 5.51 prop:lin-matrix-inverse ¶
- depends_on proposition 5.54 prop:lin-same-dim-isomorphic ¶ ↺
- depends_on proposition 5.167 Adjoint representation ¶ ↺
… 2 more

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Edges

type	direction	node	provenance	where
`depends_on`	→	Subspace generated by a set of vectors	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:840
`depends_on`	→	Linear independence	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:840
`depends_on`	←	Normal modes of a diagonalizable system	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:855
`depends_on`	←	Partial derivative; gradient	declared	parts/02-mathematical-methods/05-real-analysis.tex:2915
`depends_on`	←	Components of a vector	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:873
`depends_on`	←	Dimension	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:856
`depends_on`	←	Functional specified on a basis	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:2602
`depends_on`	←	Orthogonal basis	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1265
`depends_on`	←	The matrices of a given shape	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:691
`depends_on`	←	Exchange and completion	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1750
`depends_on`	←	The alternating top form is unique up to scale	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:5985
`depends_on`	←	Dimensions add	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:4442
`depends_on`	←	Grassmann's formula	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:4531
`depends_on`	←	prop:lin-matrix-unique	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:2073
`depends_on`	←	prop:lin-same-dim-isomorphic	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:2412
`depends_on`	←	Rank–nullity	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1829