proposition 5.45 prop:lin-matrix-unique

open in the book · parts/02-mathematical-methods/03-linear-algebra-representations.tex:2071 · p. 121

Rests on

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 4.8 Commutativity; abelian structure ¶
      - depends_on definition 4.32 Field ¶
      - depends_on definition 4.31 Module ¶
  - 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 definition 4.33 Vector space ¶ ↺
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 ¶

Supports

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 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 proposition 9.87 The distinguished limit, and which end carries the layer ¶
      - 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 9.26 Structure of the solutions ¶
- 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 ¶
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 ¶ ↺

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`	→	Basis	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:2073
`depends_on`	→	Linear transformation	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:2073
`depends_on`	→	eq:lin-indepli2	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:2073
`depends_on`	→	eq:lin-matrix-rep	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:2073
`depends_on`	←	prop:lin-matrix-algebra	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:5360
`depends_on`	←	prop:lin-matrix-inverse	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:2348
`depends_on`	←	prop:lin-same-dim-isomorphic	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:2412
`depends_on`	←	Adjoint representation	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:7822
`proves`	←	ch:03-linear-algebra-representations@proof-16	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:2075