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  • https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Abstract_Algebra%3A_Theory_and_Applications_(Judson)/12%3A_Matrix_Groups_and_Symmetry/12.02%3A_Symmetry
    \begin{align*} \| {\mathbf x} \|^2 - 2 \langle f({\mathbf x}), f({\mathbf y}) \rangle + \| {\mathbf y} \|^2 & = \| f({\mathbf x}) \|^2 - 2 \langle f({\mathbf x}), f({\mathbf y}) \rangle + \| f({\mathb...\begin{align*} \| {\mathbf x} \|^2 - 2 \langle f({\mathbf x}), f({\mathbf y}) \rangle + \| {\mathbf y} \|^2 & = \| f({\mathbf x}) \|^2 - 2 \langle f({\mathbf x}), f({\mathbf y}) \rangle + \| f({\mathbf y}) \|^2\\ & = \langle f({\mathbf x}) - f({\mathbf y}), f({\mathbf x}) - f({\mathbf y}) \rangle\\ & = \| f({\mathbf x}) - f({\mathbf y}) \|^2\\ & = \| {\mathbf x} - {\mathbf y} \|^2\\ & = \langle {\mathbf x} - {\mathbf y}, {\mathbf x} - {\mathbf y} \rangle\\ & = \| {\mathbf x} \|^2 - 2 \langle {\…

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