V is a vector space prove the intersection of two subspaces

V is a vector space. prove the intersection of two subspaces of V is a subspace of V.

Solution

To prove our statement, we will simply check that the given intersection fulfills the subspace properties stated in the definition. Let w1 and w2 be the two subspaces of V and w12 their intersection. Now we have the show the following: 1) w12 closed under addition: ----------------------------- Assume x in w12 and y in w12. From this, we know x in w1 and y in w1. But since we know that w1 is a subspace, x+y in w1 holds. Similarly, one can show that x+y in w2 and therefore x+y in w12. So w12 is indeed closed under addition. 2) w12 closed under scalar multiplication: ------------------------------------------ Assume x in w12 and r in R. Again, we know x in w1. Since w1 is a subspace, it is closed under scalar multiplication. Therefore, r*x in w1 holds. Also r*x in w2 holds with a similar argument. From this r*x in w12 follows. So w12 is closed under scalar multiplication.