U+v in subspace W, is u or v in subspace

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In summary: Therefore, if u+v is in a subspace, it does not necessarily mean that u or v is in the subspace. In other words, the subspace is not necessarily closed under addition.
  • #1
stanford1
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1. Homework Statement

My question is if u+v is in the subspace can you say that u or v is in the subspace? If not would there be a counterexample? 2. Homework Equations

closed under addition/scalar multiplication

3. The Attempt at a Solution

I know that if u or v were in the subspace they would be closed under addition or multiplication. I don't know if you can say the same for (u+v) and apply it just to u or v.

Thank you for any help.
 
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  • #2
stanford1 said:
1. Homework Statement

My question is if u+v is in the subspace can you say that u or v is in the subspace? If not would there be a counterexample?


2. Homework Equations

closed under addition/scalar multiplication

3. The Attempt at a Solution

I know that if u or v were in the subspace they would be closed under addition or multiplication. I don't know if you can say the same for (u+v) and apply it just to u or v.

Thank you for any help.

{0} is a subspace of R, a one-dimensional vector space. Are there vectors in R, that add to 0, that aren't in the subspace?

BTW, we don't talk about vectors being closed under addition or scalar multiplication - we talk about the space they belong to as being closed under addition or scalar multiplication.
 
  • #3
Just making sure I have this correctly, that would mean that a or b is not in the vector space, just a+b. Thank you for the quick response.
 
  • #4
Don't think of a + b as being two things: it's a single thing. a and b are two vectors that happen to add up to whatever value a + b represents.
 
  • #5
Mark44 said:
{0} is a subspace of R, a one-dimensional vector space. Are there vectors in R, that add to 0, that aren't in the subspace?
No, its as 0 dimension vector space. But your point is correct.

BTW, we don't talk about vectors being closed under addition or scalar multiplication - we talk about the space they belong to as being closed under addition or scalar multiplication.
 
  • #6
For example, the subset of R2, {(x, y)|y= x} is a subspace. The vectors (1, 0) and (1, 2) are not in that subspace but their sum, (2, 2), is.
 

FAQ: U+v in subspace W, is u or v in subspace

Is it possible for both u and v to be in subspace W?

Yes, it is possible for both u and v to be in subspace W. If u and v are both vectors in subspace W, then they must satisfy the properties of a subspace, such as closure under addition and scalar multiplication. As long as these properties hold, u and v can both exist in subspace W.

How can we determine if u or v is in subspace W?

To determine if u or v is in subspace W, we can check if they satisfy the properties of a subspace. This includes checking if they are closed under addition and scalar multiplication. Additionally, we can also check if they are linear combinations of other vectors that are known to be in subspace W.

What happens if u or v is not in subspace W?

If u or v is not in subspace W, then they do not satisfy the properties of a subspace. This means that they do not belong in subspace W and cannot be considered as part of the subspace. It is important to note that a vector can only belong to a subspace if it satisfies all the properties of a subspace.

Can u and v be in different subspaces within W?

Yes, u and v can be in different subspaces within W. Subspaces are defined by their own set of vectors that satisfy the properties of a subspace. As long as u and v satisfy the properties of their respective subspaces, they can exist in different subspaces within W.

How does the dimension of subspace W affect whether u or v is in subspace W?

The dimension of subspace W does not directly affect whether u or v is in subspace W. As long as u and v satisfy the properties of a subspace, they can exist in subspace W regardless of its dimension. However, the dimension of subspace W can determine the number of linearly independent vectors that can exist in the subspace, which can indirectly affect whether u or v is in subspace W.

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