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

  • Thread starter gravenewworld
  • Start date
  • Tags
    Subspace
In summary, the conversation is about finding a subset of R^2 that is closed under scalar multiplication but is not a subspace. The participants discuss the properties of a subspace and try to come up with examples that satisfy those properties. Eventually, one participant suggests the set {(x,y)|xy=0} as a solution.
  • #1
gravenewworld
1,132
26
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?
 
Physics news on Phys.org
  • #2
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?
 
  • #3
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.
 
  • #4
Hrm. Well, is there a way to take an arbitrary set and turn it into a set that is closed under multiplication?
 
  • #5
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.
 
Last edited:
  • #6
Thanks Fred. I knew it had to be something real simple that I was completely over looking.
 

1. What is subspace?

A subspace is a subset of a vector space that satisfies the properties of a vector space, such as closure under addition and scalar multiplication.

2. How do you find the U of R^2 closed under mult?

To find the set U of R^2 closed under multiplication, we need to find all possible combinations of vectors in U that result in a vector that is also in U. This can be done by checking if the product of any two vectors in U also results in a vector in U.

3. Why is it important to find the U of R^2 closed under mult?

Finding the set U of R^2 closed under multiplication is important because it helps us understand the structure of the vector space and its subspaces. It also allows us to perform operations on vectors within the subspace without leaving the subspace.

4. What does it mean for a set to be closed under multiplication?

A set is said to be closed under multiplication if the product of any two elements in the set also belongs to the set. In the context of subspaces, this means that multiplying any two vectors within the subspace will result in another vector within the subspace.

5. Can you give an example of finding U of R^2 closed under mult?

Yes, for example, if we have the set U = {(1,2),(3,4)}, we can see that the product of (1,2) and (3,4) is (3,8), which is also in the set U. Therefore, U is closed under multiplication.

Similar threads

Vector Subspaces: Determining U as a Subspace of M4x4 Matrices
    • Calculus and Beyond Homework Help
Replies
8
Views
678
I Showing that a set of differentiable functions is a subspace of R
    • Linear and Abstract Algebra
Replies
19
Views
4K
I Vector subspace and basis vectors in the context of data science
    • Linear and Abstract Algebra
Replies
6
Views
849
Constructing a Non-Subspace in R^2 with Closed Addition and Additive Inverses
    • Calculus and Beyond Homework Help
Replies
9
Views
1K
I When is a subset a subspace in a vector space?
    • Linear and Abstract Algebra
Replies
7
Views
2K
What is the induced magnetic force on this closed current loop?
    • Introductory Physics Homework Help
Replies
12
Views
187
Finding a complementary subspace ##U## | Linear Algebra
    • Calculus and Beyond Homework Help
Replies
4
Views
1K
Speed of cylinder rolling along a horizontal surface gathering snow (Solved)
    • Introductory Physics Homework Help
Replies
13
Views
877
I Prove $$T_{p}M$$ is a vector space with the axioms
    • Differential Geometry
Replies
3
Views
1K
Understanding Subspaces: Criteria and Examples (R3, Integers, R2)
    • Calculus and Beyond Homework Help
Replies
5
Views
1K

Hot Threads

Recent Insights

Back
Top