Subspaces of R3: Proof or Counterexample

  • Thread starter franky2727
  • Start date
  • Tags
    Subspace
In summary, the conversation is about determining whether certain sets are subspaces of R3. The speaker asked for help understanding the rules for determining this and whether A^2 would also be a subspace. They also questioned the relevance of certain information given in the question. The responder provided a definition and theorem for subspace and asked the speaker to check if the conditions were satisfied for the sets listed. The speaker realized that one of the sets did not contain the zero vector and questioned the others. The responder clarified that one of the conditions did not hold for one of the sets and asked the speaker to show that the other set was a subspace.
  • #1
franky2727
132
0
missed last week due to illness so have no clue what this homework is going on about, the question is for each of the following state whether or not it is a subspace of R3, Justify your answer by giving a proof or a counter example in each case, i know I'm ment to attempt the question before i ask for help but i don't know where to start, if someone could tell me what i need to look for in order to tell wheather or not it is a subspace i could then have a good at it and check up my answers after? thanks
 
Physics news on Phys.org
  • #2
done some further reading and have understood it now, however if A is a subspace is A^2 as well? can't see this anywhere in the rules just about scalars and additions
 
  • #3
also 3 parts of the question state V=((a,b,c):a,b,c E R and then the 3 questions have 3 different conditions (i) a+b=c (ii) ab=c+1 (iii) a^2=b^2 am i correct in thinking all this information is irrelivant as we have already been told that all 3 of these seperatly are E of R?
 
  • #4
franky2727 said:
done some further reading and have understood it now, however if A is a subspace is A^2 as well? can't see this anywhere in the rules just about scalars and additions
Do you mean A x A? If so, I don't think it could possibly be a subspace of the same vector space A is a subspace of.
 
  • #5
franky2727 said:
also 3 parts of the question state V=((a,b,c):a,b,c E R and then the 3 questions have 3 different conditions (i) a+b=c (ii) ab=c+1 (iii) a^2=b^2 am i correct in thinking all this information is irrelivant as we have already been told that all 3 of these seperatly are E of R?
Here are the definition of a vector subspace and a very useful theorem, from wikipedia (http://en.wikipedia.org/wiki/Vector_subspace): [Broken]
Let K be a field (such as the field of real numbers), and let V be a vector space over K. As usual, we call elements of V vectors and call elements of K scalars. Suppose that W is a subset of V. If W is a vector space itself, with the same vector space operations as V has, then it is a subspace of V.

To use this definition, we don't have to prove that all the properties of a vector space hold for W. Instead, we can prove a theorem that gives us an easier way to show that a subset of a vector space is a subspace.

Theorem: Let V be a vector space over the field K, and let W be a subset of V. Then W is a subspace if and only if it satisfies the following 3 conditions:

The zero vector, 0, is in W.
If u and v are elements of W, then the sum u + v is an element of W;
If u is an element of W and c is a scalar from K, then the scalar product cu is an element of W;​
Is the zero vector in each of the sets you listed? I can see that it is not in one of the ones you list, so that set is not a subspace (of R3). Are the other two conditions of the theorem satisfied for each set? I think I see one set for which one of the conditions doesn't hold.
 
Last edited by a moderator:
  • #6
ahh, so the zero vector isn't in the c+1 one, and i don't get what's wrong with the other two?? surely u+v is ok with the a+b=c thing? and we don't use scalars here, not sure what the rule is with squaring them tho? As for the other post again i don't get what your saying about the AxA thing? thanks for the help so far
 
  • #7
Nobody said anything is wrong with iii). You are right about what's wrong with ii). Can you show iii) is a subspace?
 

1. What is a subspace of R3?

A subspace of R3 is a subset of the three-dimensional space R3 that satisfies certain properties. It is a vector space that contains the origin, is closed under vector addition and scalar multiplication, and spans the entire space.

2. How is a subspace of R3 different from R3 itself?

R3 is the entire three-dimensional space, while a subspace of R3 is a smaller subset of it. R3 contains all possible vectors in three-dimensional space, while a subspace of R3 only contains a specific set of vectors that satisfy certain properties.

3. What are the dimensions of a subspace of R3?

A subspace of R3 can have a dimension of 0, 1, 2, or 3. A subspace of dimension 0 is simply the origin, a subspace of dimension 1 is a line passing through the origin, a subspace of dimension 2 is a plane passing through the origin, and a subspace of dimension 3 is the entire R3 space.

4. How do you determine if a set of vectors is a subspace of R3?

To determine if a set of vectors is a subspace of R3, you need to check if it satisfies the three properties of a subspace. First, the set must contain the origin. Second, it must be closed under vector addition, meaning that if you add any two vectors in the set, the result must also be in the set. Third, it must be closed under scalar multiplication, meaning that if you multiply any vector in the set by a scalar, the result must also be in the set.

5. Can a subspace of R3 contain non-numeric elements?

No, a subspace of R3 can only contain numeric elements, specifically vectors with real number components. This is because the operations of vector addition and scalar multiplication must be defined and closed within the set, and non-numeric elements would not allow for these operations to be performed.

Similar threads

  • Calculus and Beyond Homework Help
Replies
15
Views
2K
  • Calculus and Beyond Homework Help
Replies
5
Views
1K
  • Calculus and Beyond Homework Help
Replies
8
Views
1K
  • Calculus and Beyond Homework Help
Replies
4
Views
2K
  • Calculus and Beyond Homework Help
Replies
6
Views
10K
  • Calculus and Beyond Homework Help
Replies
4
Views
3K
  • Calculus and Beyond Homework Help
Replies
6
Views
1K
  • Calculus and Beyond Homework Help
Replies
4
Views
4K
  • Calculus and Beyond Homework Help
Replies
4
Views
4K
  • Calculus and Beyond Homework Help
Replies
14
Views
2K
Back
Top