# Vector Space

• Sep 3rd 2009, 08:15 PM
noppawit
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?
• Sep 3rd 2009, 08:45 PM
TKHunny
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.
• Sep 4th 2009, 04:17 AM
HallsofIvy
Quote:

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!