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/
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