I'd like to know how I can prove "the set of all vectors ∈ with and " that it is vector space or not?
By using the definition of vector space. But since you are already given that this is a set of vectors, you know that things like "associative law", etc. are true. What you need to do is see if, given that , [tex]v_3- v_4= 1[/itex], , and then the same is true of the sum . That is, is and . I recommend you look closely at that last equation!