Attachment 'devoir.tex'
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l.25 \usepackage
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l.27 \usepackage
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l.57 \begin{document}
[1]
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l.71 \begin{sagesilent}
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l.75 m = A.minimal_
polynomial()
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l.77 \end{sagesilent}
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l.77 \end{sagesilent}
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l.79 Soit $A=\sage
{A}$. Faisons quelques calculs sur la matrice $A$.
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<recently read> \sage
l.84 $\det(A)$ & $\sage
{A.det()}$ \\
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l.86 $\chi_A(x)$ & $\sage
{A.characteristic_polynomial()}$ \\
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l.88 $m_A(x)$ & $\sage
{A.minimal_polynomial()}$ \\
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l.90 Forme échelonnée & $\sage
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l.92 Diviseurs élémentaires & $\sage
{A.elementary_divisors()}$ \\
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l.97 \begin{equation*}
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l.98 m_
A(A) = \sage{m(A)}.
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l.98 m_A(A) = \sage
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l.99 \end{equation*}
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l.99 \end{equation*}
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l.112 \begin{sagesilent}
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l.114 \end{sagesilent}
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l.116 Soit maintenant $B=\sage
{B}$. Le produit de $A$ par $B$ est
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l.118 \sage
{A*B}.
[2]
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l.127 \begin{sagesilent}
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l.128 chi = A.characteristic_
polynomial()
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l.131 \end{sagesilent}
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l.131 \end{sagesilent}
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l.133 ...ire étudier la fonction $\chi_A(x)=\sage
{chi}$. Commençons par tr...
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l.133 ...de ce polynôme dans l'intervalle $[\sage
{xmin}, \sage{xmax}]$.
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l.133 ...me dans l'intervalle $[\sage{xmin}, \sage
{xmax}]$.
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l.138 \sageplot
[width=8cm]{plot(chi, xmin=xmin, xmax=xmax)}
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l.144 \begin{sageblock}
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l.145 r1,r2,r3,r4 = chi.real_
roots()
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l.146 \end{sageblock}
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l.146 \end{sageblock}
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l.148 ...s réelles de $\chi_A(x)$ qui sont $\sage
{r1}$, $\sage{r2}$, $\sage...
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l.148 ... $\chi_A(x)$ qui sont $\sage{r1}$, $\sage
{r2}$, $\sage{r3}$ et $\sa...
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l.148 ...qui sont $\sage{r1}$, $\sage{r2}$, $\sage
{r3}$ et $\sage{r4}$. Ceci...
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l.148 ...{r1}$, $\sage{r2}$, $\sage{r3}$ et $\sage
{r4}$. Ceci nous indique q...
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l.152 \sageplot
[width=8cm]{plot(chi, xmin=30, xmax=34)}
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l.161 \frac{d}{dx}\chi_A(x) = \sage
{derivative(chi)}
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l.165 \int\chi_A(x)dx = \sage
{integrate(chi)}.
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l.170 \begin{sagesilent}
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l.173 g = (x+1)^
(1/x)
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l.175 \end{sagesilent}
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l.178 \lim_{x\to 0+}\sage
{f} = \sage{limit(f, x = 0, dir='above')}
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l.178 \lim_{x\to 0+}\sage{f} = \sage
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{oo}}\sage{h} = \sage{limit(h, x = oo)}
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{h} = \sage{limit(h, x = oo)}
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{-oo}}\sage{h} = \sage{limit(h, x = -oo)}
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l.187 \lim_{x\to \sage{-oo}}\sage
{h} = \sage{limit(h, x = -oo)}
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{1/x} = \sage{limit(1/x, x = 0)}
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l.190 \lim_{x\to 0}\sage{1/x} = \sage
{limit(1/x, x = 0)}
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l.193 \lim_{x\to 0+}\sage
{1/x} = \sage{limit(1/x, x = 0, dir='above')}
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l.193 \lim_{x\to 0+}\sage{1/x} = \sage
{limit(1/x, x = 0, dir='above')}
[3]
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l.203 \begin{sagesilent}
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l.206 g = S.cayley_
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l.207 \end{sagesilent}
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l.207 \end{sagesilent}
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l.209 ...yley du groupe symétrique d'ordre $\sage
{n}$. Les couleurs des arÃ...
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l.209 ...ndent aux générateurs suivants : $\sage
{S.gens()}$.
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l.211 \sageplot
[width=6cm]{g.plot(color_by_label=True, vertex_size=0.01)}
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l.211 \sageplot[width=6cm]{g.plot(color_
by_label=True, vertex_size=0.01)}
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l.211 ...t(color_by_label=True, vertex_size=0.01)}
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l.212
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l.213 \begin{sagesilent}
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l.216 g = S.cayley_
graph()
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l.217 \end{sagesilent}
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l.217 \end{sagesilent}
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l.219 ...yley du groupe symétrique d'ordre $\sage
{n}$. Les couleurs des arÃ...
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<recently read> \sage
l.219 ...ndent aux générateurs suivants : $\sage
{S.gens()}$.
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l.221 \sageplot
[width=8cm]{g.plot(color_by_label=True, vertex_size=0.01)}
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l.221 \sageplot[width=8cm]{g.plot(color_
by_label=True, vertex_size=0.01)}
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l.221 ...t(color_by_label=True, vertex_size=0.01)}
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l.222
Overfull \hbox (132.09656pt too wide) in paragraph at lines 226--227
\OT1/cmr/m/n/12 tion disponible dans l'arborescence de sage : \OT1/cmtt/m/n/12
sage-4.1/examples/latex[]embed/sagetexpackage.pdf\OT1/cmr/m/n/12 .
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\OT1/cmtt/m/n/12 http://www.ctan.org/tex-archive/help/Catalogue/entries/sagetex
.html
[4] (./latex_7c89e2a2cdac3c3a312520e6415d658330219195_p.aux)
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(see the transcript file for additional information)
Output written on latex_7c89e2a2cdac3c3a312520e6415d658330219195_p.dvi (4 pages
, 6932 bytes).
Transcript written on latex_7c89e2a2cdac3c3a312520e6415d658330219195_p.log.
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