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The theorem of Pythagoras.

Theorem 14.2.1   \fbox{$\forall \overrightarrow{x},\overrightarrow{y} \in V,
\quad \overrightarr...
...arrow{x} \end{Vmatrix}^2 + \begin{Vmatrix}\overrightarrow{y} \end{Vmatrix}^2$ .}


\begin{proof}\par\begin{tabular}{cccccc}
$\begin{Vmatrix}\overrightarrow{x} +\ov...
...langle \overrightarrow{x} ,\overrightarrow{y}\rangle =0$\end{tabular}\end{proof}

Remark 14.2.2   The n-dimensional theorem of Pythagoras: If $\overrightarrow{x_1} , \overrightarrow{x_2} , \dots , \overrightarrow{x_n}\in V$are such that $\forall i \neq j,
\langle \overrightarrow{x_i} , \overrightarrow{x_j}\rangle =0$ (i.e. $\forall i \neq j, \overrightarrow{x_i}\perp \overrightarrow{x_j} $), then
$\begin{Vmatrix}\overrightarrow{x_1}+\overrightarrow{x_2}
+\dots +\overrightarr...
... \end{Vmatrix}^2
+ \dots + \begin{Vmatrix}\overrightarrow{x_n} \end{Vmatrix}^2$.



Noah Dana-Picard
2001-02-26