Practice FT2 Q6 - Looking for a detailed, worked solution with theoretical explanation

Suppose that Vis a real vector space of dimension 38 containing two subspaces U of dimension 12 and W of dimension 32. Let S be the intersection of U and W. Milan Pahor wants to use his mathematical magic to prove that S is a subspace of V. Help him complete his proof by selecting the options from the drop-down menus below. Proof: 1. S is a subset of the known vector space V. 2. For all x in S and y in S , x and y are both in U which is a vector space so x + y is in U Similarly, x and y are both in W which is a vector space so x + y is in W Therefore x + y is in both U and W v O and hence S This shows that S is closed under addition 3. For all A in R and x in S is in both U and W which are vectors spaces v O therefore Ax is in both U and W v and hence in This shows that S is closed under scalar multiplication Using the subspace theorem and points 1, 2 and 3, we can conclude that S is a subspace of V. Find the greatest possible dimension of S and least possible dimension of S. The greatest possible dimension of S is Number The least possible dimension of S is Number Select all of the situations below that lead to the maximum possible dimension of S O Wis the trivial vector space {0). O Wis a subset of U. O U is a subset of W. O U is the trivial vector space {0} O U and Ware the same vector space