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  • https://math.libretexts.org/Under_Construction/Purgatory/Remixer_University/Username%3A_junalyn2020/Book%3A_Introduction_to_Real_Analysis_(Lebl)/8%3A_Several_Variables_and_Partial_Derivatives/8.1%3A_Vector_Spaces%2C_linear_Mappings%2C_and_Convexity
    Let \(X\) be a set together with operations of addition, \(+ \colon X \times X \to X\), and multiplication, \(\cdot \colon {\mathbb{R}}\times X \to X\), (we write \(ax\) instead of \(a \cdot x\)). \(X...Let \(X\) be a set together with operations of addition, \(+ \colon X \times X \to X\), and multiplication, \(\cdot \colon {\mathbb{R}}\times X \to X\), (we write \(ax\) instead of \(a \cdot x\)). \(X\) is called a vector space (or a real vector space) if the following conditions are satisfied: (Addition is associative) If \(u, v, w \in X\), then \(u+(v+w) = (u+v)+w\). (Addition is commutative) If \(u, v \in X\), then \(u+v = v+u\). (Additive identity) There is a \(0 \in X\) such that \(v+0=v\)…
  • https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Abstract_Algebra%3A_Theory_and_Applications_(Judson)/12%3A_Matrix_Groups_and_Symmetry/12.01%3A_Matrix_Groups
    \begin{align*} \langle A {\mathbf x}, A {\mathbf y} \rangle & = \frac{1}{2} \left[ \|A {\mathbf x} + A {\mathbf y} \|^2 - \|A {\mathbf x} \|^2 - \|A {\mathbf y} \|^2 \right]\\ & = \frac{1}{2} \left[ \...\begin{align*} \langle A {\mathbf x}, A {\mathbf y} \rangle & = \frac{1}{2} \left[ \|A {\mathbf x} + A {\mathbf y} \|^2 - \|A {\mathbf x} \|^2 - \|A {\mathbf y} \|^2 \right]\\ & = \frac{1}{2} \left[ \|A ( {\mathbf x} + {\mathbf y} ) \|^2 - \|A {\mathbf x} \|^2 - \|A {\mathbf y} \|^2 \right]\\ & = \frac{1}{2} \left[ \|{\mathbf x} + {\mathbf y}\|^2 - \|{\mathbf x}\|^2 - \|{\mathbf y}\|^2 \right]\\ & = \langle {\mathbf x}, {\mathbf y} \rangle\text{.} \end{align*}

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