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### The Intersection of Two Subspaces

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Date: 10/11/98 at 21:43:07
From: Saul Farber
Subject: Prove that the intersection of two subspaces is a subspace

Let w1 and w2 be subspaces of the vector space V. How can I prove that
their intersection is also a subspace of V?

I have looked at the definition of a subspace, and found that it is
this:

If w1 is closed under addition and scalar multiplication (for some
r in R), then w1 is a subspace. By closed under addition, I mean
for some w and v in a subspace S, w+v is in S, by closed under
scalar multiplication, I mean for r in the reals and s in S, r*s is
also in S.

How do I prove that their intersection is a subspace? I can think
about it graphically (in Rn), and I can see why it's true, but I can't
prove it.
```

```
Date: 10/12/98 at 02:15:41
From: Doctor Anke
Subject: Re: Prove that the intersection of two subspaces is a subspace

Hello Saul,

You already know everything you need to know in order to prove that
the intersection of two subspaces is again a subspace. The definition
of a subspace is the key.

To prove our statement, we will simply check that the given
intersection fulfills the subspace properties stated in the definition.

Let w1 and w2 be the two subspaces and w12 their intersection.
Now we have the show the following:

1) w12 closed under addition:
-----------------------------
Assume x in w12 and y in w12. From this, we know x in w1 and y in w1.
But since we know that w1 is a subspace, x+y in w1 holds. Similarly,
one can show that x+y in w2 and therefore x+y in w12. So w12 is indeed

2) w12 closed under scalar multiplication:
------------------------------------------
Assume x in w12 and r in R. Again, we know x in w1. Since w1 is a
subspace, it is closed under scalar multiplication. Therefore, r*x in
w1 holds. Also r*x in w2 holds with a similar argument. From this r*x
in w12 follows. So w12 is closed under scalar multiplication.

And the proof is finished!

I hope this made things clearer. If you have further questions, feel
free to write again.

- Doctor Anke, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
College Linear Algebra
High School Linear Algebra

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