Showing that U = {(x, y) | xy ≥ 0} is not a subspace of R^2

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In summary, the task is to show that U = {(x, y) | xy ≥ 0} is not a subspace of vector space R2. The conversation discusses finding vectors that do not belong to U and whether this is sufficient to show that U is not a subspace. The conclusion is that yes, this is sufficient, as a subspace must satisfy certain properties and failing to do so means it is not a subspace.
  • #1
pakkanen
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Homework Statement


Task: Show that U = {(x, y) | xy ≥ 0} is not a subspace of vector space R2

I wish you could help me to understand why U is not a subspace of R2x2.

I have actually found a vectors u and v such that it does not belong to U (e.g. (-3,-1) +(2,2) = (-1,1) ) but is that sufficient to show that U is not a subset of R2x2?
 
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  • #2
pakkanen said:

Homework Statement


Task: Show that U = {(x, y) | xy ≥ 0} is not a subspace of vector space R2

I wish you could help me to understand why U is not a subspace of R2x2.

I have actually found a vectors u and v such that it does not belong to U (e.g. (-3,-1) +(2,2) = (-1,1) )
What does "it" refer to? are you trying to show that the sum of two vectors in the set may not be in the set? If so, yes, that is valid.

but is that sufficient to show that U is not a subset of R2x2?
Well, first, you are not trying to show U is not a subset. It is. But it is not a subspace. I suspect that was a typo. Yes, this is sufficient. To be a subspace, the subset must satisfy a number of properties. If it fails to satisfy anyone of them it is not a subspace.

Homework Statement


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The Attempt at a Solution

 
  • #3
Sorry for being unclear. I meant the subspace and not subset as you mentioned. And now it became a clear for me. Thank you very much!
 

1. What is a subspace?

A subspace is a subset of a vector space that satisfies the properties of a vector space, such as closure under vector addition and scalar multiplication.

2. How can we prove that a set is not a subspace?

To prove that a set is not a subspace, we can show that it does not satisfy one or more of the properties of a vector space.

3. What is the set U = {(x, y) | xy ≥ 0}?

The set U is the set of all ordered pairs (x, y) in the Cartesian plane where the product of x and y is greater than or equal to 0.

4. Why is U not a subspace of R^2?

U is not a subspace of R^2 because it does not satisfy the closure property under scalar multiplication. For example, if we multiply an element in U by a negative scalar, the resulting vector will not be in U.

5. Can you provide an example to show that U is not a subspace of R^2?

Yes, we can take the vectors (1, 2) and (-3, 4) which are both in U. However, if we multiply the first vector by -1, we get the vector (-1, -2) which is not in U since (-1) * (-2) = 2 is not greater than or equal to 0.

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