det_to_matrix_orthonormalBasis_real

English

The representation of basisFun recovers the coordinate map of the vector.

Русский

Представление basisFun восстанавливает координатную карту вектора.

LaTeX

$$$$ \\mathrm{basisFun}_{\\mathbb{R}}.\\mathrm{repr}(x)(i) = x_i. $$$$

Lean4

/-- The determinant of the change-of-basis matrix between two orthonormal bases `a`, `b` is ±1. -/
theorem det_to_matrix_orthonormalBasis_real : a.toBasis.det b = 1 ∨ a.toBasis.det b = -1 :=
  by
  rw [← sq_eq_one_iff]
  simpa [unitary, sq] using Matrix.det_of_mem_unitary (a.toMatrix_orthonormalBasis_mem_unitary b)

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