determine whether the following sets are subspaces of R^2 a) T={ (x1 x2) ϵ R^2 :x1x2=0}
b) V={(x1x2) ϵ│x1│=│x2│}

Question

determine whether the following sets are subspaces of R^2    a)   T={ (x1 x2) ϵ R^2 :x1x2=0}   
b) V={(x1x2) ϵ│x1│=│x2│}

jamesj · Answer

What are the axioms for a subset of a vector space to be a subspace?

Do these subsets, T and V, satisfy all of those axioms?  If so, they are subspaces.
Or do they violate at least one? In which case they are not.

If you were looking for an axiom to check first, look at closure under vector space addition.

anonymous · Answer

thnx a lot for your reply...!! yeah actually In part a I found that it is not closed under addition  but if i prove that I do not need to do something else ?? I m more stuck in exercise b ...I tried to solve it with the triangle inequality but I thing it is wrong..could you please help me ?

jamesj · Answer

the set V is also not closed under addition.

Proving that any one axiom is not satisfied is sufficient to show a subset is not a subspace.

anonymous · Answer

ok thank you very match!! for you reply again:)