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

[pakkanen]

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 R^2x2.

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 R^2x2?

 

[HallsofIvy]

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 R^2x2.

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 R^2x2?

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

Homework Equations

The Attempt at a Solution

 

[pakkanen]

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!