proposition 5.125 prop:lin-matrix-algebra

open in the book · parts/02-mathematical-methods/03-linear-algebra-representations.tex:5339 · p. 159

Rests on

depends_on definition 5.121 Algebra ¶
- 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 4.1 Cartesian product ¶
      - depends_on definition 3.43 Map ¶
  - depends_on definition 4.2 Binary operation ¶ ↺
- 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 4.9 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 5.37 Linear transformation ¶
- depends_on definition 4.33 Vector space ¶ ↺
depends_on equation 5.150 eq:lin-ctesesc2 ¶
depends_on equation 5.74 eq:lin-matrix-rep ¶
depends_on proposition 5.45 prop:lin-matrix-unique ¶
- 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 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 5.37 Linear transformation ¶ ↺
- depends_on equation 5.29 eq:lin-indepli2 ¶
- depends_on equation 5.74 eq:lin-matrix-rep ¶ ↺
- proves proof ch:03-linear-algebra-representations@proof-16 ¶
proves proof ch:03-linear-algebra-representations@proof-55 ¶

Supports

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 corollary 5.168 cor:rep-adjoint-faithful ¶ ↺
- 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 corollary 9.25 Normal modes of a diagonalizable system ¶
      - depends_on corollary 9.27 Linear stability ¶
      - depends_on proposition 9.29 The planar classification ¶
    - depends_on proposition 9.87 The distinguished limit, and which end carries the layer ¶
      - depends_on proposition 9.88 The model problem, and the error of the composite ¶
    - depends_on proposition 9.31 Lyapunov's first method: asymptotic stability ¶
    - depends_on proposition 9.43 The unmodulated line, and the feet of the tongues ¶
    - depends_on theorem 9.32 Hartman–Grobman; quoted ¶
      - depends_on theorem 32.15 Hartman–Grobman, restated from Part II ¶
    - depends_on theorem 9.26 Structure of the solutions ¶
      - depends_on corollary 9.27 Linear stability ¶ ↺
depends_on proposition 5.139 Dimension of the symplectic group ¶
- 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 remark 22.21 The count ¶

Neighborhood

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partly declared
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Edges

type	direction	node	provenance	where
`depends_on`	→	Algebra	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:5360
`depends_on`	→	Linear transformation	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:5360
`depends_on`	→	eq:lin-ctesesc2	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:5360
`depends_on`	→	eq:lin-matrix-rep	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:5360
`depends_on`	→	prop:lin-matrix-unique	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:5360
`depends_on`	←	cor:rep-adjoint-faithful	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:7869
`depends_on`	←	Representation of an algebra	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:7599
`depends_on`	←	Matrix exponential	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:738
`depends_on`	←	Dimension of the symplectic group	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6107
`proves`	←	ch:03-linear-algebra-representations@proof-55	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:5363