definition 5.12 Subspace generated by a set of vectors

open in the book · parts/02-mathematical-methods/03-linear-algebra-representations.tex:743 · p. 105

ground object -- no derivation owed

Rests on

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 4.3 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 ¶ ↺
proves proof ch:03-linear-algebra-representations@proof-3 ¶

Supports

depends_on corollary 5.108 The sum is the right operation ¶
- depends_on example 5.109 Two lines in physical space ¶
depends_on definition 5.15 Basis ¶
- 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 7.116 Functions vanishing on a regular zero set ¶
      - depends_on lemma A.510 Every diffeomorphism factorises locally ¶
      - 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 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 definition 7.127 Simple regions ¶
      - depends_on lemma A.512 The boundary strip is thin ¶
      - depends_on remark A.516 The hypotheses of the global form ¶
      - depends_on theorem 7.133 Gauss ¶
      - depends_on theorem 7.131 Green ¶
    - 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 definition A.116 Orthogonal curvilinear coordinates; scale factors ¶
      - depends_on definition A.125 Ellipsoidal coordinates ¶
      - depends_on definition A.118 Simple separation of the Helmholtz equation ¶
      - 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 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 16.15 Variation; the first variation ¶
      - 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 10.2 Linear, semilinear, quasilinear ¶
      - depends_on definition 10.24 Well-posed problem ¶
    - depends_on definition 10.66 Harmonic function ¶
      - depends_on theorem A.135 Converse of the mean-value property ¶
      - depends_on theorem A.491 Dirichlet problem for the disc; quoted ¶
      - depends_on theorem 10.69 Mean-value property ¶
      - depends_on theorem 10.75 Weak maximum principle ¶
    - depends_on definition 9.30 Hyperbolic equilibrium ¶
      - depends_on theorem 9.32 Hartman–Grobman; quoted ¶
    - depends_on lemma A.520 A $C^{1}$ limit ¶
      - depends_on proposition A.521 Every derivative passes onto $g$ ¶
    - … 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 definition 30.4 Material and spatial descriptions ¶
      - depends_on definition 31.4 Material derivative ¶
      - depends_on definition 22.26 Lagrange bracket ¶
      - depends_on definition 10.18 Complete integral; envelope ¶
      - 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 proposition 7.136 Properties of conservative fields ¶
      - 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 24.28 Ball and cylinder ¶
      - depends_on definition 24.43 Momentum map ¶
      - depends_on theorem 24.18 Symplectic form of the transformation condition ¶
    - 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 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 definition 26.12 First and second class ¶
      - depends_on proposition 26.13 The first-class functions close ¶
      - depends_on proposition 26.19 Second-class constraints come in pairs ¶
      - depends_on proposition 26.34 Proca constraints are second class ¶
    - 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 25.11 Generators of the Galilei transformations ¶
      - depends_on proposition 25.12 The algebra of generators closes ¶
    - 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 definition 24.40 Lie–Poisson bracket ¶
      - 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 notation A.636 Densities and smearing ¶
      - depends_on notation 26.36 Gauge fields and their SI dimensions ¶
    - 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 proposition 12.27 Best approximation and Bessel's inequality ¶
      - depends_on proposition 12.28 Convergence criterion for orthogonal series ¶
    - depends_on definition 5.32 Levi–Civita symbol; cross product ¶
      - depends_on definition 5.131 Isotropic Cartesian tensor ¶
      - depends_on lemma 5.34 Contraction of two Levi–Civita symbols ¶
      - depends_on lemma 5.33 Determinant through the Levi–Civita symbol ¶
    - depends_on proposition 12.23 Gram–Schmidt in a Hilbert space ¶
      - depends_on corollary 12.24 Separable spaces have countable orthonormal families ¶
      - depends_on example 12.25 Orthogonal polynomials ¶
      - depends_on proposition 12.95 The tensor inner product is well defined and positive definite ¶
    - depends_on proposition 5.28 Gram–Schmidt ¶
      - depends_on corollary 5.29 Orthogonal decomposition ¶
      - 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 proposition 5.155 Averaging trick ¶
      - depends_on theorem 5.86 Simultaneous diagonalization of a definite pencil ¶
      - depends_on theorem 5.84 Spectral theorem for a real symmetric operator ¶
- 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 proposition A.611 The constraint on $\Gamma$ ¶
    - 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 lemma A.564 Dimension and double orthogonal ¶
      - depends_on proposition A.565 Freeness makes every value regular ¶
      - 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 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 proposition 5.90 Polar decomposition ¶
      - depends_on proposition 5.130 An orthogonal transformation is an isometry ¶
      - depends_on theorem 5.71 The eigenvalues are the roots of the characteristic polynomial ¶
    - 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 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 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 proposition 5.167 Adjoint representation ¶
      - 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 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
depends_on lemma 5.38 Exchange and completion ¶ ↺

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	Linear combination	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:748
`depends_on`	→	Vector subspace	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:748
`depends_on`	←	The sum is the right operation	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:4681
`depends_on`	←	Basis	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:840
`depends_on`	←	Exchange and completion	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1750
`proves`	←	ch:03-linear-algebra-representations@proof-3	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:750