Let U and V be subspaces of Rn Prove that the intersection o

Let U and V be subspaces of Rn. Prove that the intersection of sets U and V is also a subspace of Rn.

Solution

Part I) Let x be in U intersect V. Then x is in U and x is in V (by definition of intersection).
Let y be in U intersect V. Then y is in U and y is in V (by definition of intersection).
Since x is in U and y is in U, then x+y is in U (since U is a VS and thus closed under addition).
Since x is in V and y is in V, x+y is in V (since V is a VS and thus closed under addition)
Since x+y is in U and x+y is in V, x+y is in the intersection of U and V (definition of intersection)
Part II) Let x be in U intersect V as before. Then as before, x is in U and x is in V. Let a be any scalar. Then ax is in U and ax is in V since U and V are vector spaces and thus closed under scalar multiplication. Then ax is in U intersect V (by definition of intersection).

Since U and V are vector spaces, and we have shown that U intersect V is closed under addition and scalar multiplication, U intersect V is itself a vector space, and since it is also a subset of the parent vector space R^n, it must be a subspace of R^n.