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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




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

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