The Intersection of Two Subspaces

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 
closed under addition.

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/