Question Number 146001 by puissant last updated on 10/Jul/21 $$\mathrm{Soit}\:\mathrm{p}\in\mathrm{End}\left(\mathrm{E}\right).\:\mathrm{on}\:\mathrm{pose}\:\mathrm{q}=\mathrm{id}_{\mathrm{E}} −\mathrm{p} \\ $$$$\left.\mathrm{a}\right)\:\mathrm{montrer}\:\mathrm{que}\:\mathrm{p}\:\mathrm{est}\:\mathrm{un}\:\mathrm{projecteur}\:\mathrm{si}\:\mathrm{et}\: \\ $$$$\mathrm{seulement}\:\mathrm{si}\:\mathrm{q}\:\mathrm{est}\:\mathrm{un}\:\mathrm{projecteur}.. \\ $$$$\left.\mathrm{b}\right)\:\mathrm{on}\:\mathrm{suppose}\:\mathrm{que}\:\mathrm{p}\:\mathrm{est}\:\mathrm{un}\:\mathrm{projecteur}\:\mathrm{et}\:\mathrm{on} \\ $$$$\mathrm{considere}\:\mathrm{L}=\left\{\mathrm{f}\in\mathrm{End}\left(\mathrm{E}\right)/\exists\mathrm{u}\in\mathrm{End}\left(\mathrm{E}\right),\mathrm{f}=\mathrm{u}\circ\mathrm{p}\right\} \\ $$$$\mathrm{et}\:\mathrm{M}=\left\{\mathrm{g}\in\mathrm{End}\left(\mathrm{E}\right)/\exists\mathrm{v}\in\mathrm{End}\left(\mathrm{E}\right),\:\mathrm{g}=\mathrm{v}\circ\mathrm{q}\right\}. \\ $$$$\mathrm{montrer}\:\mathrm{que}\:\mathrm{L}\:\mathrm{et}\:\mathrm{M}\:\mathrm{sont}\:\mathrm{des}\:\mathrm{sous}\:\mathrm{espaces}\: \\ $$$$\mathrm{vectoriels}\:\mathrm{supplementaires}\:\mathrm{de}\:\mathrm{End}\left(\mathrm{E}\right)..…
Question Number 146004 by puissant last updated on 10/Jul/21 $$\mathrm{F}\:\mathrm{et}\:\mathrm{G}\:\mathrm{deux}\:\mathrm{sous}\:\mathrm{espaces}\:\mathrm{vectoriels}\:\mathrm{de}\:\mathrm{E} \\ $$$$\left.\mathrm{a}\right)\:\mathrm{montrer}\:\mathrm{que}\:\left(\mathrm{F}\cap\mathrm{G}=\mathrm{F}+\mathrm{G}\right)\Leftrightarrow\left(\mathrm{F}=\mathrm{G}\right) \\ $$$$\left.\mathrm{b}\right)\:\mathrm{quand}\:\mathrm{dit}−\mathrm{on}\:\mathrm{que}\:\mathrm{les}\:\mathrm{deux}\:\mathrm{sous}\:\mathrm{espaces}\: \\ $$$$\mathrm{vectoriels}\:\mathrm{F}\:\mathrm{et}\:\mathrm{G}\:\mathrm{sont}\:\mathrm{supplementaires}? \\ $$ Answered by Olaf_Thorendsen last updated on 10/Jul/21…
Question Number 145996 by iloveisrael last updated on 10/Jul/21 $$\:\mathrm{find}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\: \\ $$$$\:\mathrm{x}−\mathrm{2x}^{\mathrm{2}} +\mathrm{3x}^{\mathrm{3}} −\mathrm{4x}^{\mathrm{4}} +\mathrm{5x}^{\mathrm{5}} −\mathrm{6x}^{\mathrm{6}} +… \\ $$$$\mathrm{where}\:\mid\mathrm{x}\mid\:<\:\mathrm{1} \\ $$ Answered by mathmax by…
Question Number 80448 by mind is power last updated on 03/Feb/20 $${Hello}\:{All}\:{of}\:{You}\:{verry}\:{Nice}\:{Day},\:{God}\:{bless}\:{You}\:{love}\:{peace}\:{and}\: \\ $$$${happiness}\: \\ $$$${Solve}\:{for}\:\left({x},{y}\right)\in\mathbb{R}^{\mathrm{2}} \: \\ $$$$\begin{cases}{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} =\mathrm{2}{x}+\mathrm{3}{y}+\mathrm{1}}\\{{x}^{\mathrm{4}} +{y}^{\mathrm{4}} =\mathrm{4}{x}^{\mathrm{2}} +\mathrm{9}{y}^{\mathrm{2}} +\mathrm{12}{xy}+\mathrm{2}{x}^{\mathrm{2}}…
Question Number 145975 by mathdanisur last updated on 09/Jul/21 $$\underset{\:\mathrm{0}} {\overset{\:\mathrm{6}} {\int}}\:\left[\:\sqrt{\mathrm{36}−{x}^{\mathrm{2}} }−\left(\mathrm{6}−{x}\right)\right]{dx}=? \\ $$ Answered by puissant last updated on 09/Jul/21 $$\mathrm{x}=\mathrm{6sin}\left(\mathrm{t}\right)\Rightarrow\mathrm{dx}=\mathrm{6cos}\left(\mathrm{t}\right)\mathrm{dt} \\ $$$$\mathrm{I}=\mathrm{6}\int_{\mathrm{0}}…
Question Number 80433 by jagoll last updated on 03/Feb/20 $${find}\:{the}\:{solution}\:{of} \\ $$$$\sqrt{\mathrm{4}−{x}}−\mathrm{2}\leqslant{x}\mid{x}−\mathrm{3}\mid+\mathrm{4}{x} \\ $$ Commented by john santu last updated on 03/Feb/20 $$\left(\mathrm{1}\right)\:\mathrm{4}−{x}\geqslant\mathrm{0}\:\Rightarrow{x}\leqslant\mathrm{4} \\ $$$$\left(\mathrm{2}\right)\sqrt{\mathrm{4}−{x}\:}\leqslant{x}\mid{x}−\mathrm{3}\mid+\mathrm{4}{x}+\mathrm{2}…
Question Number 145953 by mathdanisur last updated on 09/Jul/21 Commented by mathdanisur last updated on 09/Jul/21 $${Solve}\:{the}\:{trigonometric}\:{equation} \\ $$ Commented by puissant last updated on…
Question Number 145954 by mathdanisur last updated on 09/Jul/21 $$\mathrm{1}+{i}+{i}^{\mathrm{2}} +{i}^{\mathrm{3}} +…+{i}^{\mathrm{99}} =? \\ $$ Answered by mr W last updated on 09/Jul/21 $$=\frac{\mathrm{1}−{i}^{\mathrm{100}} }{\mathrm{1}−{i}}=\frac{\mathrm{1}−\left(−\mathrm{1}\right)^{\mathrm{50}}…
Question Number 145947 by Khalmohmmad last updated on 09/Jul/21 Commented by hknkrc46 last updated on 09/Jul/21 $$\:\left.\begin{matrix}{\int\boldsymbol{{f}}\left(\boldsymbol{{x}}\right)\boldsymbol{{dx}}\:=\:\boldsymbol{{F}}\left(\boldsymbol{{x}}\right)\:+\:\boldsymbol{{c}}}\\{\int\boldsymbol{{g}}\left(\boldsymbol{{x}}\right)\boldsymbol{{dx}}\:=\:\boldsymbol{{G}}\left(\boldsymbol{{x}}\right)\:+\:\boldsymbol{{c}}}\end{matrix}\right\}\:\begin{matrix}{\boldsymbol{{G}}\left(\mathrm{2}\right)\:=\:\mathrm{3}}\\{\boldsymbol{{f}}\left(\mathrm{5}\right)\:=\:\mathrm{1}}\end{matrix} \\ $$$$ \\ $$$$\:\:\:\bullet\:\boldsymbol{{f}}\left(\boldsymbol{{x}}\right)\:=\:\boldsymbol{{F}}\:'\left(\boldsymbol{{x}}\right)\:\wedge\:\boldsymbol{{g}}\left(\boldsymbol{{x}}\right)\:=\:\boldsymbol{{G}}\:'\left(\boldsymbol{{x}}\right) \\ $$$$\:\:\:\bullet\:\frac{\boldsymbol{{d}}\left[\left(\boldsymbol{{x}}^{\mathrm{2}} \:+\:\mathrm{2}\right)\boldsymbol{{G}}\left(\boldsymbol{{x}}\right)\right]}{\boldsymbol{{dx}}}\:=\:\frac{\boldsymbol{{d}}\left[\boldsymbol{{F}}\left(\mathrm{3}\boldsymbol{{x}}\:−\:\mathrm{1}\right)\right]}{\boldsymbol{{dx}}} \\…
Question Number 80404 by peter frank last updated on 02/Feb/20 Answered by MJS last updated on 02/Feb/20 $${f}\left({x}\right):\:{y}=\frac{{x}^{\mathrm{2}} }{\mathrm{5}}\:\Rightarrow\:{f}^{−\mathrm{1}} \left({y}\right):\:{x}=\sqrt{\mathrm{5}{y}} \\ $$$${x}'=\frac{\sqrt{\mathrm{5}}}{\mathrm{2}\sqrt{{y}}} \\ $$$$\mathrm{surface}=\mathrm{2}\pi\underset{{a}} {\overset{{b}}…