Treatise of Physics

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lemma 5.137 The alternating top form is unique up to scale

open in the book · parts/02-mathematical-methods/03-linear-algebra-representations.tex:5967 · p. 166

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lemma 5.137: The alternating top form is unique up to scale5.137definition 5.15: Basis5.15definition 5.110: Bilinear map5.110equation 5.19: eq:lin-leibniz-det5.19theorem 5.138: A symplectic transformation has determinant +15.138proof : ch:03-linear-algebra-representations@proof-60proofdefinition 5.12: Subspace generated by a set of vectors5.12definition 5.14: Linear independence5.14corollary 9.25: Normal modes of a diagonalizable system9.25definition 7.97: Partial derivative; gradient7.97definition 5.17: Components of a vector5.17definition 5.16: Dimension5.16definition 5.60: Functional specified on a basis5.60definition 5.26: Orthogonal basis5.26example 5.9: The matrices of a given shape5.9lemma 5.38: Exchange and completion5.38proposition 5.102: Dimensions add5.102proposition 5.105: Grassmann's formula5.105proposition 5.45: prop:lin-matrix-unique5.45proposition 5.54: prop:lin-same-dim-isomorphic5.54theorem 5.40: Rank–nullity5.40definition 5.37: Linear transformation5.37definition 12.94: Tensor product of Hilbert spaces12.94definition 5.113: Non-degenerate form5.113definition 5.111: Symmetric and antisymmetric forms5.111proposition 5.114: prop:lin-bilinear-dual5.114proposition 5.112: Symmetric–antisymmetric splitting5.112proposition 5.119: Normal form of a non-degenerate antisymmetric form5.119theorem 5.116: Sylvester's law of inertia5.116definition A.508: Primitive mapA.508definition 5.58: Special linear group5.58definition 5.70: Characteristic polynomial5.70lemma A.497: The determinant is multiplicativeA.497lemma A.507: Coordinate permutationsA.507lemma 5.33: Determinant through the Levi–Civita symbol5.33proposition 7.111: Jacobi's formula, cofactor form7.111proposition 5.74: prop:lin-geometric-le-algebraic5.74proposition 5.2: Jacobi's formula, column form5.2proposition 13.6: Levi-Civita identities in three dimensions13.6definition 5.135: The symplectic group5.135neighborhood truncated

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typedirectionnode provenancewhere
depends_on → Basis declared parts/02-mathematical-methods/03-linear-algebra-representations.tex:5985
depends_on → Bilinear map declared parts/02-mathematical-methods/03-linear-algebra-representations.tex:5985
depends_on → eq:lin-leibniz-det declared parts/02-mathematical-methods/03-linear-algebra-representations.tex:5985
depends_on ← A symplectic transformation has determinant $+1$ declared parts/02-mathematical-methods/03-linear-algebra-representations.tex:6036
proves ← ch:03-linear-algebra-representations@proof-60 declared parts/02-mathematical-methods/03-linear-algebra-representations.tex:5988