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Linear Algebra: Direct sum proof

by loesch.19
Tags: direct sum, linear algebra
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loesch.19
#1
May31-10, 04:43 PM
P: 3
Let U and V be subspaces of a vector space W. If W=U [tex]\oplus[/tex] V, show U [tex]\bigcap[/tex] V={0}.

I'm a bit lost on this one... as I thought this was essentially the definition of direct sum. I'm unsure where to start. Any help would be great!
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vela
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May31-10, 06:16 PM
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What's the definition of a direct sum you're using?
loesch.19
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May31-10, 07:52 PM
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Quote Quote by vela View Post
What's the definition of a direct sum you're using?
I wasn't sure if that was necessary info or not... looks like I was wrong :)

If U and V are subspaces of vector space W, and each w in W can be written uniquely as a sum u+v where u is in U and v is in V then W is a direct sum of U and V.

Hurkyl
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May31-10, 08:00 PM
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Linear Algebra: Direct sum proof

So were you able to do the problem?
loesch.19
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May31-10, 08:04 PM
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Quote Quote by Hurkyl View Post
So were you able to do the problem?
No... I still need help.
Hurkyl
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May31-10, 08:12 PM
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Well, surely you can do something on it -- even if it's just rewriting the problem in a less opaque form.

e.g. do you know anything about proving two subspaces equal? (Or two sets?)
Susanne217
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Jun1-10, 06:03 AM
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Quote Quote by loesch.19 View Post
Let U and V be subspaces of a vector space W. If W=U [tex]\oplus[/tex] V, show U [tex]\bigcap[/tex] V={0}.

I'm a bit lost on this one... as I thought this was essentially the definition of direct sum. I'm unsure where to start. Any help would be great!
I think you should add the definition as follows:

If U and V are subspaces of a Vector space W and each [tex]w \in W[/tex] can be written as the unique sum as u+v where [tex]u \in U[/tex] and [tex]v \in V[/tex] then

W is the direct sum of U and V and can be written [tex]W = U \oplus V[/tex]
HallsofIvy
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Jun1-10, 06:41 AM
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Suppose there were a non-zero vector, w, in both U and V and let u be any vector in U. Now, write u as two different sums of vectors in U and V.


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