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Oscar Plaisant
2024-09-20 21:31:23 +02:00
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construction de C.md
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@@ -16,7 +16,7 @@ $\left( \begin{array}{cc}a&-b\\b&a\end{array} \right)$ avec $(a, b)\in\mathbb R^
Alors toute matrice $A$ s'écrit :
$\begin{aligned} \left( \begin{array}{cc}a&-b\\b&a\end{array} \right) &= \left( \begin{array}{cc}a&0\\0&a\end{array} \right) +\left( \begin{array}{cc}0&-b\\b&0\end{array} \right) \\[.5em] &= a\left( \begin{array}{cc}1&0\\0&1\end{array} \right) + \left( \begin{array}{cc}0&-1\\1&0\end{array} \right)b \\[.5em] &= a\text{Id} + b\text{Im} \end{aligned}$
$\begin{aligned} \left( \begin{array}{cc}a&-b\\b&a\end{array} \right) &= \left( \begin{array}{cc}a&0\\0&a\end{array} \right) +\left( \begin{array}{cc}0&-b\\b&0\end{array} \right) \\[.5em] &= a\left( \begin{array}{cc}1&0\\0&1\end{array} \right) + b \left( \begin{array}{cc}0&-1\\1&0\end{array} \right) \\[.5em] &= a\text{Id} + b\text{Im} \end{aligned}$
On peut alors construire une [[bijection]] avec $\mathbb C$, et même un [[isomorphisme]]. Cela permet de construire $\mathbb C$.