Question Number 134009 by mathocean1 last updated on 26/Feb/21
![E is a vec torial space which has as base B=(i^→ ,j^→ ,k^→ ). f: E→E is a linear application such that f(i^→ )=−i^→ +2k^→ ; f(j^→ )=j^→ +2k^→ and j(k^→ )=2i^→ +2j^→ . 1. Write the matrix of f in base B. 2. Show that the kernel (ker f) of f is a straigh line; give one base of its. 3.Determinate Im f.](https://www.tinkutara.com/question/Q134009.png)
$${E}\:{is}\:{a}\:{vec}\:{torial}\:{space}\:{which}\:{has}\:{as} \\ $$$${base}\:\mathscr{B}=\left(\overset{\rightarrow} {{i}},\overset{\rightarrow} {{j}},\overset{\rightarrow} {{k}}\right).\:{f}:\:{E}\rightarrow{E}\:{is}\:{a}\:{linear} \\ $$$${application}\:{such}\:{that} \\ $$$${f}\left(\overset{\rightarrow} {{i}}\right)=−\overset{\rightarrow} {{i}}+\mathrm{2}\overset{\rightarrow} {{k}};\:{f}\left(\overset{\rightarrow} {{j}}\right)=\overset{\rightarrow} {{j}}+\mathrm{2}\overset{\rightarrow} {{k}}\:{and} \\ $$$${j}\left(\overset{\rightarrow} {{k}}\right)=\mathrm{2}\overset{\rightarrow} {{i}}+\mathrm{2}\overset{\rightarrow} {{j}}. \\ $$$$\mathrm{1}.\:\boldsymbol{{W}}{rite}\:{the}\:{matrix}\:{of}\:{f}\:{in}\:{base}\:\mathscr{B}. \\ $$$$\mathrm{2}.\:{Show}\:{that}\:{the}\:{kernel}\:\left({ker}\:{f}\right)\:{of}\:{f} \\ $$$${is}\:{a}\:{straigh}\:{line};\:{give}\:{one}\:{base}\:{of}\:{its}. \\ $$$$\mathrm{3}.{Determinate}\:{Im}\:{f}. \\ $$