Help with Subspace: Find U of R^2 Closed Under Mult.

[gravenewworld]

I have been trying this problem for hours. I can't believe I can't get it. The question is "Find a subset U of R^2 such that U is closed under scalar multiplication but is not a subspace of R^2". I know that for U to be a subspace 0 must be an element of U and U has to be closed under scalar multiplication and vector addition. If U is closed under scalar multiplication then it must contain O vector right? So I have to think of a subset that is not closed under addition, but is closed under multiplication right? I can not think of any. Does anyone have an idea of one that satifies these properties?

 

[Hurkyl]

Can you think of any subsets of R^2 that aren't closed under addition?

Study your examlpes of sets closed under multiplication. Can you identify any properties they have in common? Can you then try to come up with a new example without that property?

 

[gravenewworld]

I can think of subsets of R^2 that aren't closed under addition, but the fact that the set is closed under scalar multiplicaton always messes things up for me. For example {(x1,x2): x1,x2 don't=0}. I know that this isn't closed under addition because (x1,x2)+(-x1,-x2)=0 vector. But this also isn't closed under scalar multiplicaiton because 0 is an element of the field R, and 0(x1,x2)=(0,0). I have thought of examples that aren't closed under addition, but they always violate the fact that they must be closed under multiplication.

 

[Hurkyl]

Hrm. Well, is there a way to take an arbitrary set and turn it into a set that is closed under multiplication?

 

[Fredrik]

The set {(x,y)|xy=0}, satisfies the required properties. This is just the set you suggested plus the element (0,0).

In order to be closed under scalar multiplication, the set has to be a union of straight lines passing through (0,0). The only proper subsets of that kind that are closed under addition are the ones that consist of only one such line.

 

[gravenewworld]

Thanks Fred. I knew it had to be something real simple that I was completely over looking.