I am really confused by this vector subspace question:
Determine whether each given set S is a subspace of the given vector space V.
V = R2, and S = {(x,y) -> V | x <= y}
I concluded that (0,0) works for zero (1,1) and (-1,2) added = (0,3) works for addition (-1,2) x 2 = (-2,4) works for multiplication because if you multiply a smaller number and a larger number by the same scalar then the smaller number will still be smaller and the larger number will still be larger.
The answer I was given states that this does not pass by multiplication with no examples or proof so I'm kind of lost because this seems straightforward to me. I can't find a combination that this doesn't work for.
Looking for some validation or an explanation before I address this with the professor, thanks.
This set does not pass the scalar multiplication test. Consider the vector $(1,2)$, which is an element of $S$. If I multiply this vector by $-1$, I get $(-1,-2)$, which is not an element of $S$. It's multiplication by negative numbers that causes the issue.