proposition 5.139 Dimension of the symplectic group

open in the book · parts/02-mathematical-methods/03-linear-algebra-representations.tex:6094 · p. 168

Rests on

depends_on definition 5.127 Lie algebra ¶
- 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 4.2 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 4.20 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 5.135 The symplectic group ¶
- depends_on definition 5.113 Non-degenerate form ¶
  - depends_on definition 5.110 Bilinear map ¶
    - depends_on definition 5.37 Linear transformation ¶
      - depends_on definition 4.33 Vector space ¶ ↺
- depends_on equation 5.138 eq:lin-bilinear-expansion ¶
- depends_on proposition 5.119 Normal form of a non-degenerate antisymmetric form ¶
  - depends_on definition 5.110 Bilinear map ¶ ↺
  - depends_on definition 5.113 Non-degenerate form ¶ ↺
  - depends_on definition 5.111 Symmetric and antisymmetric forms ¶
    - depends_on definition 5.110 Bilinear map ¶ ↺
    - depends_on equation 5.137 eq:lin-bilinear-matrix ¶
  - depends_on equation 5.139 eq:lin-bilinear-congruence ¶
  - proves proof ch:03-linear-algebra-representations@proof-54 ¶
depends_on proposition 5.125 prop:lin-matrix-algebra ¶
- depends_on definition 5.121 Algebra ¶ ↺
- depends_on definition 5.37 Linear transformation ¶ ↺
- 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 5.7 Vector subspace ¶
      - 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 ¶
proves proof ch:03-linear-algebra-representations@proof-62 ¶

Supports

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

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`	→	Lie algebra	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6107
`depends_on`	→	The symplectic group	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6107
`depends_on`	→	prop:lin-matrix-algebra	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6107
`depends_on`	←	Properties of the symplectic group	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:165
`depends_on`	←	The count	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:711
`proves`	←	ch:03-linear-algebra-representations@proof-62	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6111