Question Number 97380 by mathocean1 last updated on 07/Jun/20
$${E}\:{is}\:{a}\:{vectorial}\:{plane}.\:{his}\:{base}\:{is}\: \\ $$$${B}=\left(\overset{\rightarrow} {{i}};\overset{\rightarrow} {{j}}\right).\:{f}\:{is}\:{an}\:{endomorphism}\:{defined} \\ $$$${by}\:{f}\left(\overset{\rightarrow} {{i}}\right)=−\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\overset{\rightarrow} {{i}}+\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\overset{\rightarrow} {{j}}\:{and}\:{f}\left(\overset{\rightarrow} {{j}}\right)=\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\overset{\rightarrow} {{i}}−\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\overset{\rightarrow} {{j}} \\ $$$$\left.\mathrm{1}\right){Show}\:{that}\:{ker}\:{f}\:{is}\:{a}\:{vectorial}\:{straigh} \\ $$$${line}\:{and}\:{his}\:{base}\:{is}\:\overset{\rightarrow} {{e}}_{\mathrm{1}} =\sqrt{\mathrm{2}}\overset{\rightarrow} {{i}}+\sqrt{\mathrm{2}}\overset{\rightarrow} {{j}} \\ $$$$\left.\mathrm{2}\right){show}\:{that}\:{G},\:{the}\:{set}\:{of}\:{vectors}\:\overset{\rightarrow} {{u}} \\ $$$$\:\in\:{E}\:{such}\:{as}\:{f}\left(\overset{\rightarrow} {{u}}\right)=\sqrt{\mathrm{2}}\overset{\rightarrow} {{u}}\:{is}\:{a}\:{vectorial}\:{straigh} \\ $$$${line}\:{and}\:{his}\:{Base}\:{is}\:\overset{\rightarrow} {{e}}_{\mathrm{2}\:\:} =\overset{\rightarrow} {{i}}+\overset{\rightarrow} {{j}} \\ $$$$\left.\mathrm{3}\right)\:{Determine}\:{the}\:{matrix}\:{A}'\:{of}\:{f}\:{in} \\ $$$${B}'\:{if}\:{B}'=\left(\overset{\rightarrow} {{e}}_{\mathrm{1}} ;\overset{\rightarrow} {{e}}_{\mathrm{2}} \right). \\ $$