Linear Algebra: Direct sum proof

In summary: One way will use u itself and the other will use w.Since each w \in W can be written uniquely as u+v, the two representations you've written for u must be the same. But, since one of them uses w and the other doesn't, we can cancel w from both sides. What's left?In summary, if U and V are subspaces of a vector space W and W is the direct sum of U and V, then their intersection is {0}.
  • #1
loesch.19
3
0
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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  • #2
What's the definition of a direct sum you're using?
 
  • #3
vela said:
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.
 
  • #4
So were you able to do the problem?
 
  • #5
Hurkyl said:
So were you able to do the problem?

No... I still need help.
 
  • #6
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?)
 
  • #7
loesch.19 said:
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]
 
  • #8
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.
 

What is a direct sum in linear algebra?

A direct sum in linear algebra is a way of combining two subspaces of a vector space in a way that preserves their individual structures. It is represented by the symbol ⊕ and is denoted as the direct sum of two subspaces V and W as V ⊕ W.

What is the difference between direct sum and direct product in linear algebra?

The direct sum and direct product are similar concepts, but they differ in how they combine subspaces. In a direct sum, the two subspaces are combined in a way that preserves their individual structures, while in a direct product, the two subspaces are combined in a way that creates a new structure that is a combination of the two.

How do you prove that two subspaces form a direct sum in linear algebra?

To prove that two subspaces V and W form a direct sum, you must show that their intersection is the zero vector and that the sum of their dimensions is equal to the dimension of their sum, or V + W. This can be done using a proof by contradiction or by using the definition of a direct sum.

What are the applications of direct sums in linear algebra?

Direct sums have various applications in linear algebra, including in the study of vector spaces, linear transformations, and matrix algebra. They are also used in areas such as coding theory, cryptography, and signal processing.

Can more than two subspaces form a direct sum in linear algebra?

Yes, more than two subspaces can form a direct sum in linear algebra. In fact, any finite collection of subspaces can form a direct sum if their intersection is the zero vector and the sum of their dimensions is equal to the dimension of their sum.

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