I need some help with these proofs.
1. Prove that the intersection of any collection of subspaces of V is a subspace of V.
2. Suppose that U is a subspace of V. What is U + U ?
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 .