Question: Let S, T and U be subspaces of a vector space V. We can form new subspaces using the operations of and + defined

Let S, T and U be subspaces of a vector space V. We can form new subspaces using the operations of ∩ and + defined in Exercises 18 and 20. When we do arithmetic with numbers, we know that the operation of multiplication distributes over the operation of addition in the sense that
a(b + c) = ab + ac
It is natural to ask whether similar distributive laws hold for the two operations with subspaces.
(a) Does the intersection operation for subspaces distribute over the addition operation; that is, does
S ∩ (T + U) = (S ∩ T) + (S ∩ U)
(b) Does the addition operation for subspaces distribute over the intersection operation; that is, does
S + (T ∩ U) = (S + T) ∩ (S + U)

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