Answer to Question #320972 in Linear Algebra for Talifhani

Question

Prove that if U and V are subspaces of Rn

so is U+V

Answer 1

$U+V=\left\{ u+v,u\in U,v\in V \right\} \\We\,\,need\,\,to\,\,check\,\,that\,\,\forall \alpha ,\beta \in \mathbb{R} \,\,\forall x,y\in U+V\\\alpha x+\beta y\in U+V\\We\,\,have\\x=u_1+v_1,u_1\in U,v_1\in V\\y=u_2+v_2,u_2\in U,v_2\in V\\\alpha x+\beta y=\alpha \left( u_1+v_1 \right) +\beta \left( u_2+v_2 \right) =\left( \alpha u_1+\beta u_2 \right) +\left( \alpha v_1+\beta v_2 \right) \\Since\,\,U,V\,\,are\,\,subspaces, \alpha u_1+\beta u_2\in U,\alpha v_1+\beta v_2\in V\Rightarrow \\\Rightarrow \alpha x+\beta y\in U+V, which\,\,was\,\,to\,\,be\,\,proved$