How to show U and W direct sum of V?

In summary, the conversation discusses showing that V=R^3 is the direct sum of W and U, where W is generated by (1, 0, 0) and U is generated by (1, 1, 0) and (0, 1, 1). The objective is to prove that each vector in V can be expressed as a unique sum of a vector in U and a vector in W. The first step is to show that a specific vector can be expressed in this way, and then to show that this decomposition is unique. This can be done by showing that the vectors in U and W are linearly independent, and since there are 3 vectors, they span all of R^3.
  • #1
cookiesyum
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Let V = R^3. Let W be the space generated by w = (1, 0, 0), and let U be the subspace generated by u_1 = (1, 1, 0) and u_2 = (0, 1, 1). Show that V is the direct sum of W and U.
 
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  • #2
Try and follow the forum guidelines, ok? Show an attempt or state what's confusing you. Don't just post the question. You want to show each vector in V=R^3 can be expressed as the sum of a vector in U and a vector in W in a unique way. Try it.
 
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  • #3
Dick said:
Try and follow the forum guidelines, ok? Show an attempt or state what's confusing you. Don't just post the question. You want to show each vector in V=R^3 can be expressed as the sum of a vector in U and a vector in W in a unique way. Try it.

Yes, I will. I'm sorry about the misunderstanding, I'm new to this forum.

First I let v = (2, 2, 1) E in R^3. Then I showed that v = u1 + u2 + w = (1 1 0) + (0 1 1) + (1 0 0) = (2 2 1) which still E in R^3. Now, how do I show that the intersection of U and W contains only 0. Do I do this by showing u1, u2, and w are linearly independent?
 
  • #4
Well, you've only shown that one particular vector can be expressed as a sum, but you have show that way of decomposing v is unique. But, yes, if you show u1, u2 and w are linearly independent than you know that the coefficients are unique, and since there are 3 vectors, they span all of R^3.
 

1. What is the definition of direct sum in linear algebra?

In linear algebra, the direct sum of two vector spaces V and W is a third vector space that contains all possible combinations of vectors from V and W. In other words, it is a way of combining two vector spaces without overlapping any of their elements.

2. How do you show that U and W are direct sum of V?

To show that U and W are direct sum of V, you need to prove that their sum is equal to V and their intersection is equal to the zero vector. This means that every vector in V can be written as a unique combination of vectors from U and W, and U and W do not share any common vectors.

3. What are the conditions for U and W to be direct sum of V?

There are two conditions that must be met for U and W to be direct sum of V. Firstly, the sum of U and W must be equal to V. Secondly, the intersection of U and W must be equal to the zero vector. If both of these conditions are satisfied, then U and W are direct sum of V.

4. Can a vector space have more than two direct summands?

Yes, a vector space can have more than two direct summands. In general, a vector space can have any number of direct summands as long as the sum of all the direct summands is equal to the original vector space.

5. How can you prove that two vector spaces are direct sum of each other?

To prove that two vector spaces are direct sum of each other, you need to show that their sum is equal to the original vector space and their intersection is equal to the zero vector. This can be done by finding a basis for each vector space and showing that they are linearly independent and span the original vector space.

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