Question

Let U and V be subspaces of R^n. Prove that U intersect V is also a subspace of R^n

Answer 1

Proof. We will use the subspace test. Let u,v ∈ U∩V, and let r be any scalar. Then u and v are each contained in both U and V . Since U is closed under addition, u + v ∈ U. Since V is closed under addition, u+v∈V. Therefore u+v∈U∩V, so U∩V is closed under addition.
Since U is closed under scalar multiplication, ru ∈ U. Since V is closed under scalar multiplication, ru ∈ V . Therefore ru ∈ U ∩ V , so U ∩ V is closed under scalar multiplication.
Therefore U ∩ V is a subspace of R^n .