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 ?

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- Jan 22nd 2007, 02:12 PMtaypezLinear 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 ? - Jan 22nd 2007, 02:23 PMThePerfectHacker
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 . - Jan 22nd 2007, 02:31 PMtopsquark
:eek: If I may just say that has to be the most interesting reason for deleting a message I've ever seen...

-Dan - Jan 22nd 2007, 02:45 PMtaypez
2) Makes sense, but this problem comes before Direct Sums. Would there be another proof perhaps using just sums?

- Feb 17th 2009, 05:59 PMcookiesyum