definition 5.5 Linear combination

open in the book · parts/02-mathematical-methods/03-linear-algebra-representations.tex:581 · p. 103

ground object -- no derivation owed

Rests on

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 4.2 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 4.3 Algebraic structure ¶ ↺
  - depends_on definition 4.20 Monoid ¶
    - depends_on definition 4.9 Neutral element ¶ ↺
    - depends_on definition 4.18 Semigroup ¶
      - depends_on definition 4.6 Associativity ¶
      - depends_on definition 4.4 Internal binary operation; magma ¶ ↺
  - depends_on definition 4.30 Ring ¶
    - depends_on definition 4.8 Commutativity; abelian structure ¶ ↺
    - depends_on definition 4.28 Distributivity ¶
      - depends_on definition 4.3 Algebraic structure ¶ ↺
    - depends_on definition 4.21 Group ¶
      - depends_on definition 4.11 Inverse element ¶ ↺
      - depends_on definition 4.20 Monoid ¶ ↺
    - 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 ¶ ↺

Supports

depends_on definition 5.101 External direct sum ¶
- depends_on definition 12.84 External direct sum ¶
  - depends_on proposition 12.85 The direct sum is a Hilbert space ¶
    - depends_on proposition 12.87 Expansion in an orthogonal decomposition ¶
      - depends_on lemma A.250 Decomposition into cyclic subspaces ¶
- depends_on definition 5.103 Sum of subspaces; internal direct sum ¶
  - depends_on corollary 5.108 The sum is the right operation ¶
    - depends_on example 5.109 Two lines in physical space ¶
  - depends_on proposition 5.104 Criterion for a direct sum ¶
  - depends_on proposition 5.105 Grassmann's formula ¶
- depends_on proposition 5.102 Dimensions add ¶
  - depends_on proposition 5.104 Criterion for a direct sum ¶ ↺
depends_on definition 5.12 Subspace generated by a set of vectors ¶
- depends_on corollary 5.108 The sum is the right operation ¶ ↺
- 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 9.31 Lyapunov's first method: asymptotic stability ¶
      - depends_on proposition 9.29 The planar classification ¶
      - depends_on theorem 9.39 Floquet ¶
    - 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 7.119 Envelope of a family ¶
      - depends_on definition 7.126 Line and surface integrals ¶
      - depends_on definition 7.127 Simple regions ¶
      - depends_on definition A.508 Primitive map ¶
      - depends_on definition A.116 Orthogonal curvilinear coordinates; scale factors ¶
      - depends_on definition A.298 Half-space; smoothness on it ¶
      - depends_on definition 16.11 Admissible class; functional ¶
      - depends_on definition 10.1 Partial differential equation; order ¶
      - depends_on definition 10.66 Harmonic function ¶
      - depends_on definition 9.30 Hyperbolic equilibrium ¶
      - depends_on lemma A.520 A $C^{1}$ limit ¶
      - … 8 more
    - depends_on definition 7.102 Differential of a function ¶
      - depends_on proposition 7.104 Chain rule in several variables ¶
      - 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 lemma A.75 Differentiating a pullback along a flow ¶
      - 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 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 26.12 First and second class ¶
      - depends_on definition 26.8 Total Hamiltonian ¶
      - 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 notation 26.28 Field systems ¶
      - depends_on notation 26.1 Weak and strong equality ¶
      - depends_on proposition 24.15 Poisson bracket from the 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 5.32 Levi–Civita symbol; cross product ¶
      - depends_on proposition 12.23 Gram–Schmidt in a Hilbert space ¶
      - depends_on proposition 5.28 Gram–Schmidt ¶
  - 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 5.74 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 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 ¶
  - 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 corollary 24.19 Canonical transformations preserve phase volume ¶
      - depends_on proposition 24.5 Properties of the symplectic 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 9.21 Matrix exponential ¶
      - depends_on proposition 5.139 Dimension of the symplectic group ¶
    - 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 ¶
      - depends_on corollary 5.168 cor:rep-adjoint-faithful ¶ ↺
  - … 2 more
- depends_on lemma 5.38 Exchange and completion ¶ ↺
depends_on definition 5.14 Linear independence ¶
- depends_on corollary 5.62 A functional that annihilates a set of constraints ¶ ↺
- depends_on definition 5.15 Basis ¶ ↺
- depends_on lemma 5.38 Exchange and completion ¶ ↺
- depends_on lemma 13.136 Discrete subgroups of $\R^{f}$ ¶
- 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 proposition 5.31 Gram criterion ¶
- depends_on proposition 5.28 Gram–Schmidt ¶ ↺
- depends_on proposition 5.20 Cauchy–Schwarz inequality ¶
  - depends_on proposition 12.4 Cauchy–Schwarz and continuity of the inner product ¶
    - depends_on corollary 12.5 Continuity of the norm and of orthogonality ¶
      - depends_on proposition 12.17 The complement is always a closed subspace ¶
    - depends_on lemma A.606 $F$ is entire, and is the overlap in disguise ¶
      - depends_on lemma A.607 A zero-free entire function is an exponential ¶
    - depends_on lemma A.254 Riemann integral of a continuous curve ¶
      - depends_on lemma A.255 Smoothed vectors lie in the domain ¶
      - depends_on lemma A.257 Integrated form of the equation of motion ¶
      - depends_on lemma A.583 Absolutely convergent operator-valued integrals ¶
    - depends_on lemma A.583 Absolutely convergent operator-valued integrals ¶ ↺
    - depends_on proposition 12.102 Neither plane waves nor deltas are in $L^{2}$ ¶
      - depends_on example 12.104 The Schwartz triple ¶
    - depends_on theorem 25.53 Properties of the Wigner function ¶
      - depends_on lemma A.603 Overlaps and the resolution of unity ¶
depends_on definition 5.7 Vector subspace ¶
- depends_on corollary 5.29 Orthogonal decomposition ¶
  - depends_on proposition 5.153 prop:rep-unitary-completely-reducible ¶
    - depends_on theorem 101.62 The mixing matrix is unitary ¶
      - depends_on proposition 101.63 Parameter counting, and why three generations permit $CP$ violation ¶
      - depends_on theorem 103.11 GIM suppression at one loop ¶
  - depends_on theorem 5.43 The four fundamental subspaces ¶ ↺
- depends_on definition 12.13 Convex set ¶
  - depends_on theorem 12.14 Closest point in a closed convex set ¶
    - depends_on theorem 12.18 Projection theorem ¶
      - depends_on corollary 12.19 Double complement; the density criterion ¶
      - depends_on definition A.248 Cyclic vector and cyclic subspace ¶
      - depends_on definition 12.20 Orthogonal projection operator ¶
      - depends_on lemma A.233 Construction of the system ¶
      - depends_on lemma A.231 Restriction to an invariant closed subspace ¶
      - depends_on lemma A.267 Isometry of $A\pm\ii\mu$, and closed range ¶
      - depends_on proposition 12.27 Best approximation and Bessel's inequality ¶
      - depends_on proposition 12.21 Characterization of orthogonal projections ¶
      - depends_on theorem 12.46 Riesz representation ¶
- depends_on definition 5.12 Subspace generated by a set of vectors ¶ ↺
- depends_on definition 5.103 Sum of subspaces; internal direct sum ¶ ↺
- depends_on definition 5.39 Kernel, image, nullity, rank ¶
  - depends_on definition 5.69 Eigenvector, eigenvalue, eigenspace ¶
    - depends_on corollary 9.25 Normal modes of a diagonalizable system ¶ ↺
    - 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 proposition 5.75 Eigenvectors for distinct eigenvalues are independent ¶ ↺
    - 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 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 lemma 5.97 Fitting splitting ¶ ↺
  - depends_on proposition 5.49 Injective, surjective, invertible ¶ ↺
  - depends_on theorem 5.40 Rank–nullity ¶ ↺
- 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 corollary 5.168 cor:rep-adjoint-faithful ¶ ↺
- depends_on definition 5.147 Invariant subspace ¶
  - depends_on definition 5.148 Irreducible representation ¶
    - depends_on proposition 5.161 Irreducible representations of an abelian group ¶
    - depends_on proposition 5.151 prop:rep-not-totally-reducible ¶
    - depends_on proposition 5.153 prop:rep-unitary-completely-reducible ¶ ↺
    - depends_on theorem 5.158 Schur's first lemma ¶
      - depends_on corollary 14.18 A Casimir acts as a number on an irreducible representation ¶
      - depends_on lemma A.427 Invariant bilinear forms on a simple algebra ¶
      - depends_on lemma A.405 Schur, algebra form ¶
      - depends_on phenomenon 102.1 Hadron masses obey an octet mass formula ¶
      - depends_on proposition 5.161 Irreducible representations of an abelian group ¶ ↺
      - depends_on theorem 106.72 Confinement at strong coupling ¶
      - depends_on theorem 102.12 Which quark combinations can be colour singlets ¶
    - depends_on theorem 5.160 Schur's second lemma ¶
      - depends_on lemma 25.39 The low-degree images are forced ¶
      - depends_on theorem 25.38 Groenewold–van Hove ¶
  - depends_on definition 5.149 Totally reducible representation ¶
    - depends_on proposition 5.151 prop:rep-not-totally-reducible ¶ ↺
    - depends_on proposition 5.153 prop:rep-unitary-completely-reducible ¶ ↺
  - depends_on lemma 5.150 Invariance of the orthogonal complement ¶
    - depends_on proposition 5.151 prop:rep-not-totally-reducible ¶ ↺
    - depends_on proposition 5.153 prop:rep-unitary-completely-reducible ¶ ↺
  - depends_on theorem 5.158 Schur's first lemma ¶ ↺
  - depends_on theorem 5.160 Schur's second lemma ¶ ↺
- depends_on example 5.10 Functions on a set ¶
- depends_on proposition 5.61 Annihilator of a subspace ¶ ↺
- depends_on proposition 5.107 The union of two subspaces ¶
  - depends_on corollary 5.108 The sum is the right operation ¶ ↺
  - depends_on example 5.109 Two lines in physical space ¶ ↺
depends_on example 5.8 The coordinate spaces $\R^{n}$ and $\C^{n}$ ¶
depends_on proposition 5.107 The union of two subspaces ¶ ↺

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	Vector space	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:589
`depends_on`	←	External direct sum	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:4427
`depends_on`	←	Subspace generated by a set of vectors	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:748
`depends_on`	←	Linear independence	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:816
`depends_on`	←	Vector subspace	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:611
`depends_on`	←	The coordinate spaces $\R^{n}$ and $\C^{n}$	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:667
`depends_on`	←	The union of two subspaces	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:4650