1. ## Linear Algebra II

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 ?

2. Originally Posted by taypez
I need some help with these proofs.

1. Prove that the intersection of any collection of subspaces of V is a subspace of V.
The subspace must contain $\bold{0}$ thus their intersection is non-empty.

We need to show vector addition closure and scalar multiplication closure.
Let $\b{u},\b{v} \in W_1\cap W_2$
Then,
$\b{u}+\b{v} \in W_1$ because it is a subspace since $\b{u},\b{v}\in W_1$.
And,
$\b{u}+\b{v} \in W_2$ because it is a subspace since $\b{u},\b{v}\in W_2$.
Thus,
$\b{u}+\b{v}\in W_1\cap W_2$.

Next, let $k$ be any scalar.
Then,
$k\b{u}\in W_1$ because it is a subspace and $\b{u}\in W_1$.
And,
$k\b{u}\in W_2$ because it is a subspace and $\b{u}\in W_2$.
Thus,
$k\b{u}\in W_1\cap W_2$.

Originally Posted by taypez
2. Suppose that U is a subspace of V. What is U + U ?
I presume that means the direct product (or direct sum).
Meaning,
$U+U=\{(x,y)|x,y\in U\}$.
I seems that,
$U+U$ will be a subspace $V+V$.

3. If I may just say that has to be the most interesting reason for deleting a message I've ever seen...

-Dan

4. 2) Makes sense, but this problem comes before Direct Sums. Would there be another proof perhaps using just sums?

5. Originally Posted by ThePerfectHacker
The subspace must contain $\bold{0}$ thus their intersection is non-empty.

We need to show vector addition closure and scalar multiplication closure.
Let $\b{u},\b{v} \in W_1\cap W_2$
Then,
$\b{u}+\b{v} \in W_1$ because it is a subspace since $\b{u},\b{v}\in W_1$.
And,
$\b{u}+\b{v} \in W_2$ because it is a subspace since $\b{u},\b{v}\in W_2$.
Thus,
$\b{u}+\b{v}\in W_1\cap W_2$.

Next, let $k$ be any scalar.
Then,
$k\b{u}\in W_1$ because it is a subspace and $\b{u}\in W_1$.
And,
$k\b{u}\in W_2$ because it is a subspace and $\b{u}\in W_2$.
Thus,
$k\b{u}\in W_1\cap W_2$.

I presume that means the direct product (or direct sum).
Meaning,
$U+U=\{(x,y)|x,y\in U\}$.
I seems that,
$U+U$ will be a subspace $V+V$.
What would U + V be then? It is a space, but is it a subspace of V?