Question Number 134009 by mathocean1 last updated on 26/Feb/21
$${E}\:{is}\:{a}\:{vec}\:{torial}\:{space}\:{which}\:{has}\:{as} \\ $$$${base}\:\mathscr{B}=\left(\overset{\rightarrow} {{i}},\overset{\rightarrow} {{j}},\overset{\rightarrow} {{k}}\right).\:{f}:\:{E}\rightarrow{E}\:{is}\:{a}\:{linear} \\ $$$${application}\:{such}\:{that} \\ $$$${f}\left(\overset{\rightarrow} {{i}}\right)=−\overset{\rightarrow} {{i}}+\mathrm{2}\overset{\rightarrow} {{k}};\:{f}\left(\overset{\rightarrow} {{j}}\right)=\overset{\rightarrow} {{j}}+\mathrm{2}\overset{\rightarrow} {{k}}\:{and} \\ $$$${j}\left(\overset{\rightarrow} {{k}}\right)=\mathrm{2}\overset{\rightarrow} {{i}}+\mathrm{2}\overset{\rightarrow} {{j}}. \\ $$$$\mathrm{1}.\:\boldsymbol{{W}}{rite}\:{the}\:{matrix}\:{of}\:{f}\:{in}\:{base}\:\mathscr{B}. \\ $$$$\mathrm{2}.\:{Show}\:{that}\:{the}\:{kernel}\:\left({ker}\:{f}\right)\:{of}\:{f} \\ $$$${is}\:{a}\:{straigh}\:{line};\:{give}\:{one}\:{base}\:{of}\:{its}. \\ $$$$\mathrm{3}.{Determinate}\:{Im}\:{f}. \\ $$