Sums of Subspaces: Is Addition Commutative & Associative?

[bjgawp]

If $$U_1, U_2, U_3,$$ are subspaces of V (over fields R and/or C), is the addition of the subspaces commutative and associative?

To me it seems rather trivial .. Since their summation is simply the set of all possible sums of the elements of $$U_1, U_2, U_3$$, and the elements themselves are associative and commutative, then so must be their subspaces and their sum.

Seems too easy to me ... I must be missing something ,,

 

[g_edgar]

It is true. What you wrote is fine. But if you are doubtful, then you need to write it out more carefully.