Question

Let U and W be subspaces of some vector space V. Prove that U∩W is a subspace of V

Answer 1

To prove that the intersection U∩W of two subspaces U and W is a subspace of V, one must show it contains the zero vector, is closed under vector addition, and is closed under scalar multiplication, all of which are true for U∩W.

Step-by-step explanation:

A student has inquired if the intersection of two subspaces U and W of a vector space V is itself a subspace of V. To prove that U∩W is a subspace, we need to verify that it meets three criteria: (1) the zero vector of V is in U∩W, (2) U∩W is closed under vector addition, and (3) U∩W is closed under scalar multiplication.

Proof

1. The zero vector: Since U and W are subspaces, they must each contain the zero vector. The zero vector is also in their intersection U∩W.
2. Closure under addition: Let a and b be elements of U∩W. Since they are in U and W, their sum a + b is in both U and W, so a + b must be in U∩W.
3. Closure under scalar multiplication: For a scalar c and an element a of U∩W, the product c⋅a is in both U and W because they are subspaces, hence c⋅a is in U∩W

These points satisfy the necessary and sufficient conditions for U∩W to be a subspace of V.