ThmDex – An index of mathematical definitions, results, and conjectures.
Euclidean real dot product is symmetric
Formulation 0
Let $x, y \in \mathbb{R}^{N \times 1}$ each be a D5200: Real column matrix.
Then \begin{equation} x \cdot y = y \cdot x \end{equation}
Formulation 1
Let $x, y \in \mathbb{R}^{N \times 1}$ each be a D5200: Real column matrix.
Then \begin{equation} x^T y = y^T x \end{equation}
Formulation 2
Let $x, y \in \mathbb{R}^{N \times 1}$ each be a D5200: Real column matrix.
Then \begin{equation} \sum_{n = 1}^N x_n y_n = \sum_{n = 1}^N y_n x_n \end{equation}
Proofs
Proof 0
Let $x, y \in \mathbb{R}^{N \times 1}$ each be a D5200: Real column matrix.