1. ## Vector Space

I'd like to know how I can prove "the set of all vectors $\displaystyle V = [v_1, v_2, v_3, v_4]$∈$\displaystyle R^4$ with $\displaystyle v_1+v_2=0$ and $\displaystyle v_3-v_4 = 1$" that it is vector space or not?

2. Not really the right place for that kind of algebra.

Look through your book. There is a numbered list of requirements to be a Vector Space. Find it and prove it - one requirement at a time.

3. Originally Posted by noppawit
I'd like to know how I can prove "the set of all vectors $\displaystyle V = [v_1, v_2, v_3, v_4]$∈$\displaystyle R^4$ with $\displaystyle v_1+v_2=0$ and $\displaystyle v_3-v_4 = 1$" 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 $\displaystyle v_1+ v_2= 0$, [tex]v_3- v_4= 1[/itex], $\displaystyle u_1+ u_2= 0$, and $\displaystyle u_3- u_4= 1$ then the same is true of the sum $\displaystyle [v_1, v_2, v_3, v_4]+ [u_1, u_2, u_3, u_4]= [v_1+u_1, v_2+ u2, v_3+ u_3, v_4+u_4]$. That is, is $\displaystyle (v_1+ u_1)+ (v_2+ u_2)= 0$ and $\displaystyle (v_3+ u_3)- (v_4+ u_4)= 1$. I recommend you look closely at that last equation!