Why is the following not a subspace?

[pyroknife]

I attached the problem, the solutions say its not a subspace.

To be a subspace it must satisfy 3 conditions
1) 0 is in S
2) if U and V are in S, then U+V must be in S
3) if V is in S, then fV is in S for some scalar f.


0 is in S

U+V is in S because if U and V have elements that are negative for y, then the addition or those are still less than 0.

fV isn't in S because if the scalar was a negative number then y would be greater than 0. Is this why it's not a subspace?

 

[haruspex]

3) if V is in S, then fV is in S for some scalar f.

That should be "for any scalar f"

fV isn't in S because if the scalar was a negative number then y would be greater than 0. Is this why it's not a subspace?

Yes, but I'd prefer to be a little more precise. You should say "fV is not always in S", and it won't generate an exception if y happens to be zero. Give a specific example V, f, for which fV would not be in S.

 

[hedipaldi]

Exactly.It is also not closed under summation.