The Tangle of Western Mathematics regarding Isometry.
Pythagoras → Descartes → Euler/Lagrange/Gauss → Grassmann/Hamilton → Cayley/Sylvester → Gram/Schmidt → modern linear algebra. PYTHAGOREAN THEOREM c² = a² + b² │ │ establishes the special relationship │ between perpendicular components and length ▼ ORTHOGONAL COORDINATE COMPONENTS v = (x₁,x₂,...,xₙ) │ │ generalizes Pythagoras ▼ EUCLIDEAN NORM ||v||² = x₁² + x₂² + ... + xₙ² │ │ can be written ▼ DOT PRODUCT v·v = ||v||² │ │ and for two vectors ▼ ORTHOGONALITY u·v = 0 │ │ allows an orthonormal basis ▼ column-stacking vectors → column matrix → Gram matrix MᵀM → orthonormal set → orthonormal basis → square Q → orthogonal matrix → orthogonal group O(N). CARTESIAN / ORTHONORMAL BASIS eᵢ·eⱼ = δᵢⱼ │ │ transform the basis: │ e₁→q₁, e₂→q₂, ... eₙ→qₙ ▼ MATRIX Q = [q₁ | q₂ | ... | qₙ] │ │ ask whether the new basis │ retains the old dot products ▼ GRAM MATRIX QᵀQ │ │ if those dot products remain │ exactly those of the Cartes...