plx help....Let U be a vector space, U x U the product vector space, and
# = { (u,u)|u in U} ⊆ U x U
Prove that # is a subspace of U x U and that (U x U) / # (isomorphism) U.
plx help....Let U be a vector space, U x U the product vector space, and
# = { (u,u)|u in U} ⊆ U x U
Prove that # is a subspace of U x U and that (U x U) / # (isomorphism) U.
What about the obvious choice of $\displaystyle T:U\boxplus U\to Uu_1,u_2)\to u_1-u_2$. This is evidently surjective since $\displaystyle T_{\mid U\boxplus\{\bold{0}\}}=\text{id}_{U}$ and it's a homomorphism since $\displaystyle T(\alpha(u_1,u_2)+\beta(u_3,u_4))=\left(\alpha u_1+\beta u_3,\alpha u_2+\beta u_4)=(\alpha u_1+\beta u_3)-(\alpha u_2+\beta u_4)=\alpha(u_1-u_2)+\beta(u_3-u_4)=\alpha T((u_1,u_2))+\beta T((u)3,u_4))$. I claim that we are now finished. Why?