Chapter 3.2, Problem 26E

Let S, T, and U be subspaces of a vector space V. We can form new subspaces using the operations of $\cap$ and + defined in Exercises 23 and 25. When we do arithmetic with numbers, we know that the operation of multiplication distributes over the operation of addition in the sense that
$a\left(b+c\right)=ab+ac$
It is natural to ask whether similar distributive laws hold for the two operations with subspaces.
(a) Does the intersection operation foe subspaces distribute over the addition operation? That is, does
$S\cap \left(T+U\right)=\left(S\cap T\right)+\left(S\cap U\right)$
(b) Does the addition operation for subspaces distribute over the intersection operation That is, does
$S+\left(T+U\right)=\left(S+T\right)\cap \left(S+U\right)$