Prove that if U and V are subspaces of Rn
so is U+V
U+V={u+v,u∈U,v∈V}We need to check that ∀α,β∈R ∀x,y∈U+Vαx+βy∈U+VWe havex=u1+v1,u1∈U,v1∈Vy=u2+v2,u2∈U,v2∈Vαx+βy=α(u1+v1)+β(u2+v2)=(αu1+βu2)+(αv1+βv2)Since U,V are subspaces,αu1+βu2∈U,αv1+βv2∈V⇒⇒αx+βy∈U+V,which was to be provedU+V=\left\{ u+v,u\in U,v\in V \right\} \\We\,\,need\,\,to\,\,check\,\,that\,\,\forall \alpha ,\beta \in \mathbb{R} \,\,\forall x,y\in U+V\\\alpha x+\beta y\in U+V\\We\,\,have\\x=u_1+v_1,u_1\in U,v_1\in V\\y=u_2+v_2,u_2\in U,v_2\in V\\\alpha x+\beta y=\alpha \left( u_1+v_1 \right) +\beta \left( u_2+v_2 \right) =\left( \alpha u_1+\beta u_2 \right) +\left( \alpha v_1+\beta v_2 \right) \\Since\,\,U,V\,\,are\,\,subspaces, \alpha u_1+\beta u_2\in U,\alpha v_1+\beta v_2\in V\Rightarrow \\\Rightarrow \alpha x+\beta y\in U+V, which\,\,was\,\,to\,\,be\,\,provedU+V={u+v,u∈U,v∈V}Weneedtocheckthat∀α,β∈R∀x,y∈U+Vαx+βy∈U+VWehavex=u1+v1,u1∈U,v1∈Vy=u2+v2,u2∈U,v2∈Vαx+βy=α(u1+v1)+β(u2+v2)=(αu1+βu2)+(αv1+βv2)SinceU,Varesubspaces,αu1+βu2∈U,αv1+βv2∈V⇒⇒αx+βy∈U+V,whichwastobeproved
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