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

## 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?

## Answers and Replies

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HallsofIvy
Science Advisor
Homework Helper

## 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 any one of them it is not a subspace.

## The Attempt at a Solution

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!