Give an example of an Fvector space V and subspaces U1 U2 of

Give an example of an F-vector space V and subspaces U_1, U_2 of V such that U_1 times U_2 is isomorphic to U_1 + U_2, but U_1 + U_2 is not a direct sum.

Solution

Product and direct sum of finitely many subspaces are the same things.

So, product isomorphic to the usual sum of subspaces, automatically implies the sum is actually direct sum.

Therefore, the question above is contradictory. It\'s not possible.