# Is this a vector space if addition is defined by…?

$\mathbb{R}^2$, with the real scalar multiplication but addition defined by $$\begin{bmatrix}x_1 \\ y_1\end{bmatrix}+\begin{bmatrix}x_2 \\ y_2\end{bmatrix}=\begin{bmatrix}x_1+x_2+1 \\ y_1+y_2+1\end{bmatrix}$$

why or why not? I'm confused about how to go about this problem. I know the conditions that they must fulfill, but the specific addition definition is confusing me..

Hint: Check whether the scalar multiple $0v$ of some vector$~v$ is the neutral element for addition (this has to be true in a vector space for any vector $v$).
You can go about this by checking if there exists a neutral element of the addition, i.e., a vector $z\in\mathbb R^2$ such that $x+z=x$ for all $x\in\mathbb R^2$. Just try any two vectors, e.g., $(1, 0)^T$ and $(1,1)^T$ and see if such $z$ can exist.