Let

and

be two subspaces of a vector space

over the field

. Prove that the intersection of

and

is a subspace of

.

Can someone check my working for this question

Let

Since

is a subspace, it is closed under addition and so

Similarly, let

Since

is a subspace, it is closed under addition and so

Closure under addition

Let

since

is a subspace

since

is a subspace

Clearly,

Closure under scalar multiplication

is a subspace of

What about the condition for non-empty? This condition always confuses me. Why is it that the zero vector has to be included for it to be non-empty?