# Is the following in $\mathbb{R}^2$ a vector space?

Is the following in $\mathbb{R}^2$ a vector space?

$u=[x_1,y_1],v=[x_2,y_2]$

Addition defined as follows: $u+v=[x_1+x_2+1,y_1+y_2]$

Multiplication defined as $ru=[rx+r-1,ry], r\in\mathbb{R}$

One of the axioms say that there exists an element in V, denote by $0$ such that v + $0$=v

Does this $0$ have to be the $0$ vector $[0,0]$?

Because in this case, $v+\text{ 0 vector}=[x_2+1,y_2]\neq v$

So does this make is not a vector space? Or can $0=[-1, 0]$, in which case it would work?

• No, $0$ doesn't have to be $(0,0)$. Sep 10, 2017 at 17:44
• There is something up with your indices. $y_2$ is a component in both $u$ and $v$, and are you certain that your addition works the way it does? Because as it stands, the first component of $u+v$ only cares about $u$ and the second component only about $v$. Sep 10, 2017 at 17:52
• Sorry it is a mistake typing I will fix Sep 10, 2017 at 18:09

$0$ has to be the vector satisfying $v+0=v$ for all vectors $v$ in your space. If your vector addition is the standard vector addition, then $0$ is $(0,0)$. If your addition is not the standard addition, then $0$ might be something else.