Understanding Equivalence Relations in Real Numbers and Vector Spaces

[Kate2010]

Homework Statement

I have got myself very confused about equivalence relations. I have to determine whether certain relations R are equivalence relations (and if they are describe the partition into equivalence classes, but I'll worry about that once I understand the first part).

Here are some of the relations:

i) S = {x is a real number| x>0}; aRb iff ab = 1

ii) S = R^3\ {0}; vRw iff there is a 1 dimensional subspace of the vector space R^3 which contains v and w.

I have several other relations to consider but is someone could give me some ideas with these, hopefully I will be able to transfer my understanding to the others.

Many thanks.

Homework Equations

The Attempt at a Solution

i) S = {x is a real number| x>0}; aRb iff ab = 1

If my understanding is correct, this is not an equivalence relation as aRa is not true for all a in S, e.g. if a = 2 then 2*2=4 is not equal to 1, so R is not reflexive.

ii) S = R^3\ {0}; vRw iff there is a 1 dimensional subspace of the vector space R^3 which contains v and w.

I have no idea on this one.

 

[vela]

The Attempt at a Solution

i) S = {x is a real number| x>0}; aRb iff ab = 1

If my understanding is correct, this is not an equivalence relation as aRa is not true for all a in S, e.g. if a = 2 then 2*2=4 is not equal to 1, so R is not reflexive.

Yes, that's correct.

ii) S = R^3\ {0}; vRw iff there is a 1 dimensional subspace of the vector space R^3 which contains v and w.

I have no idea on this one.

Hint: A one-dimensional subspace has a basis containing just one vector, so the subspace is just the set of all multiples of that vector.

 

[Kate2010]

Ok, then I think it is an equivalence relation. Would the equivalence classes be [v] = {av} where a is in R?

 

[vela]

Yup, though a can't be 0 since the zero vector isn't in S.