Sums of Subspaces: Is Addition Commutative & Associative?

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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 ,,
 
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