Subspaces of R3: Proof or Counterexample

[franky2727]

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

 

[franky2727]

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

 

[franky2727]

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?

 

[Mark44]

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.

 

[Mark44]

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):

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.

 

[franky2727]

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

 

[Dick]

Nobody said anything is wrong with iii). You are right about what's wrong with ii). Can you show iii) is a subspace?