**Chapter 1 Exercise 8.**

Prove the intersection of any collection of subspaces of *V* is a subspace of *V* .

*Proof*:

Let be subspaces of .

Then . Thus,

(1)

Therefore, .

Suppose . Then .

Since is a subspace, . Thus

(2)

Since is a subspace, and scalar . Thus

(3)

By 1, 2 and 3, is a subspace of .