The subspace must contain thus their intersection is non-empty.

We need to show vector addition closure and scalar multiplication closure.

Let

Then,

because it is a subspace since .

And,

because it is a subspace since .

Thus,

.

Next, let be any scalar.

Then,

because it is a subspace and .

And,

because it is a subspace and .

Thus,

.

I presume that means the direct product (or direct sum).

Meaning,

.

I seems that,

will be a subspace .