Question Number 79111 by mathocean1 last updated on 22/Jan/20
$$\mathrm{Show}\:\mathrm{that} \\ $$$$\mathrm{E}=\left\{\left({x},\mathrm{y},{z}\right)\:\in\:\mathbb{R}^{\mathrm{3}} \:\:/\:\:{x}−\mathrm{2}{y}+{z}=\mathrm{0}\right\} \\ $$$$\mathrm{is}\:\mathrm{a}\:\mathrm{subspace}\:\mathrm{vector}\:\mathrm{of}\:\mathrm{which}\:\mathrm{we} \\ $$$$\mathrm{will}\:\mathrm{determine}\:\mathrm{one}\:\mathrm{base}. \\ $$$$\mathrm{please}\:\mathrm{help}\:\mathrm{sirs}... \\ $$
Commented by mathmax by abdo last updated on 22/Jan/20
$${x}−\mathrm{2}{y}\:+{z}\:=\mathrm{0}\:\Rightarrow{x}=\mathrm{2}{y}−{z}\:\Rightarrow\left({x},{y},{z}\right)=\left(\mathrm{2}{y}−{z},{y},{z}\right) \\ $$$$=\left(\mathrm{2}{y},{y},{o}\right)\:+\left(−{z},\mathrm{0},{z}\right)\:={y}\left(\mathrm{2},\mathrm{1},\mathrm{0}\right)\:+{z}\left(−\mathrm{1},\mathrm{0},\mathrm{1}\right)\:={y}\:\overset{\rightarrow} {{u}}\:+{z}\overset{\rightarrow} {{v}}\:\Rightarrow \\ $$$${E}\:{is}\:{a}\:{vectorial}\:{plane}\:{with}\:{base}\:{B}=\left(\overset{\rightarrow} {{u}},\overset{\rightarrow} {{v}}\right) \\ $$$$\overset{\rightarrow} {{u}}\left(\mathrm{2},\mathrm{1},\mathrm{0}\right)\:{and}\:\overset{\rightarrow} {{v}}\left(−\mathrm{1},\mathrm{0},\mathrm{1}\right) \\ $$
Commented by mathocean1 last updated on 22/Jan/20
$$\mathrm{thanks}\:\mathrm{sir} \\ $$
Commented by mathmax by abdo last updated on 23/Jan/20
$${you}\:{are}\:{welcome} \\ $$