Question Number 86021 by mathocean1 last updated on 26/Mar/20
$${E}\:{is}\:{a}\:{vectorial}\:{plan}\:{in}\:\mathbb{R}\:{with}\:{a}\:{base} \\ $$$${B}=\left(\overset{\rightarrow} {{i}},\overset{\rightarrow} {{j}}\right).\:{f}\:{is}\:{an}\:{endomorphism}\:{of}\:{E} \\ $$$${defined}\:\forall\:\overset{\rightarrow} {{u}}={x}\overset{\rightarrow} {{i}}+{y}\overset{\rightarrow} {{j}}\:{by}\:{f}\left(\overset{\rightarrow} {{u}}\right)=\left(−\mathrm{7}{x}−\mathrm{12}{y}\right)\overset{\rightarrow} {{i}}+\left(\mathrm{4}{x}+\mathrm{7}{y}\right)\overset{\rightarrow} {{j}}. \\ $$$$\left.\mathrm{1}\right)\:{Determinate}\:{f}\left(\overset{\rightarrow} {{i}}\right)\:{and}\:{f}\left(\overset{\rightarrow} {{j}}\right)\:\:{then}\: \\ $$$${write}\:{the}\:{matrice}\:{of}\:{f}\:{in}\:\left(\overset{\rightarrow} {{i}},\overset{\rightarrow} {{j}}\right){base}. \\ $$
Commented by mathocean1 last updated on 27/Mar/20
$$\left.\mathrm{2}\right)\:{The}\:{following}\:{question}\:{is}: \\ $$$${Determinate}\:{the}\:{matrice}\:{of}\:{g}={f}\mathrm{o}{f} \\ $$$${i}\:{found}\:\begin{pmatrix}{\mathrm{1}\:\:\:\:\:\:\:\:\mathrm{0}}\\{\mathrm{0}\:\:\:\:\:\:\:\:\mathrm{1}}\end{pmatrix} \\ $$$$\left.\mathrm{3}\right){show}\:{that}\:{g}\left(\overset{\rightarrow} {{i}}\right)=\overset{\rightarrow\:} {{i}}\:\:\:{and}\:{g}\left(\overset{\rightarrow} {{j}}\right)=\overset{\rightarrow} {{j}} \\ $$$${i}\:{showed}\:{it}. \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:{f}\mathrm{o}{f}\left(\overset{\rightarrow} {{u}}\right)… \\ $$$${Can}\:{you}\:{help}\:{me}\:{for}\:{this}\:{please}… \\ $$$$… \\ $$$$… \\ $$$$… \\ $$$$\left.\mathrm{5}\right)\:{E}_{\mathrm{1}} =\left\{\overset{\rightarrow} {{u}}\:\in\:{E}/\:{f}\left(\overset{\rightarrow} {{u}}\right)=\overset{\rightarrow} {{u}}\right\}\: \\ $$$$\:\:\:\:\:\:{E}_{\mathrm{2}} =\left\{\overset{\rightarrow\:} {{u}}\in\:{E}/{f}\left(\overset{\rightarrow} {{u}}\right)=−\overset{\rightarrow} {{u}}\right\} \\ $$$$\bullet{Show}\:{that}\:{E}_{\mathrm{1}} \:{and}\:{E}_{\mathrm{2}} \:\:{are}\:{vectorial} \\ $$$${lines}. \\ $$$$\bullet{then}\:{give}\:{one}\:{of}\:{theirs}\:{bases}\:\overset{\rightarrow} {{e}}_{\mathrm{1}\:} {and}\:\overset{\rightarrow} {{e}}_{\mathrm{2}} . \\ $$$$\bullet{show}\:{that}\:\left(\overset{\rightarrow} {{e}}_{\mathrm{1}} ,\overset{\rightarrow} {{e}}_{\mathrm{2}} \right)\:{is}\:{a}\:{base}\:{of}\:{E}. \\ $$$$\ast{Determinate}\:{the}\:{matrice}\:{of}\:{f}\:{in}\:{this} \\ $$$${base}. \\ $$$$ \\ $$$${Please}\:{i}\:{need}\:{your}\:{help}\:{sirs}… \\ $$
Commented by mathocean1 last updated on 26/Mar/20
$${can}\:{you}\:{help}\:{me}\:{please}… \\ $$
Commented by abdomathmax last updated on 26/Mar/20
$${i}\left(\mathrm{1},\mathrm{0}\right)\:\Rightarrow{f}\left({i}\right)\:=−\mathrm{7}{i}\:+\mathrm{4}{j}\:\:\Rightarrow{f}\left({i}\right)\begin{pmatrix}{−\mathrm{7}}\\{\mathrm{4}}\end{pmatrix} \\ $$$${j}\left(\mathrm{0},\mathrm{1}\right)\:\Rightarrow{f}\left({j}\right)\:=−\mathrm{12}{i}\:+\mathrm{7}{j}\:\Rightarrow{f}\left({j}\right)\begin{pmatrix}{−\mathrm{12}}\\{\mathrm{7}}\end{pmatrix} \\ $$$${M}_{{f}} \left({i},{j}\right)\:=\begin{pmatrix}{−\mathrm{7}\:\:\:\:\:\:\:\:\:−\mathrm{12}}\\{\mathrm{4}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{7}}\end{pmatrix} \\ $$