线性代数验证题求解

(u1,v1)+(u2,v2)=(u1+u2,v1+v2) belongs to W ( because u1+u2 belongs to U and v1+v2 belongs to V ); a(u,v)=(au,av) belongs to W ( because au belongs to U and av belongs to V );(u1,v1)+(u2,v2)=(u1+u2,v1+v2)=(u2+u1,v2+v1)=(u2,v2)+(u1,v1);((u1,v1)+(u2,v2))+(u3,v3)=(u1+u2,v1+v2)+(u3,v3)=((u1+u2)+u3,(v1+v2)+v3)=(u1+(u2+u3),v1+(v2+v3))=(u1,v1)+(u2+u3,v2+v3)=(u1,v1)+((u2,v2)+(u3,v3));(01,02) is the zero element,here 01 is the zero element of U and 02 is the zero element of V;For a element (u,v) in W,the element (-u,-v) is the negative element,here -u is the negative element of u in U and -v is the negative element of v in V;1*(u,v)=(1*u,1*v)=(u,v);k(l(u,v))=k(lu,lv)=(k(lu)。k(lv))=((kl)u,(kl)v)=(kl)(u,v);(k+l)(u,v)=((k+l)u,(k+l)v)=(ku+lu,kv+lv)=(ku,kv)+(lu,lv)=k(u,v)+l(u,v);k((u1,v1)+(u2,v2))=k(u1+u2,v1+v2)=(ku1+ku2,kv1+kv2)=(ku1,kv1)+(ku2,kv2)=k(u1,v1)+k(u2,v2)。