Q:

Let $V$ be the set of all ordered pairs of real numbers with addition defined by $(x_1,x_2)+(y_1,y_2)=(x_1+y_1,x_2+y_2)$

Let $V$ be the set of all ordered pairs of real numbers with addition defined by
$$(x_1,x_2)+(y_1,y_2)=(x_1+y_1,x_2+y_2)$$
and scalar multiplication defined by
$$\alpha\circ(x_1,x_2)=(\alpha x_1,x_2)$$
Is $V$ a vector space with these operations?

A:

If $V$ is a vector space then, for any $v\in V$, $0v=(0+0)v=0v+0v\implies 0v=0$.
Your definition of scalar multiplication is that $a(x_1,x_2)=(ax_1,x_2)$.
Therefore, $0(1,2)=(0,2)\ne (0,0)$ so $V$ is not a vector space.
edm pointed out that I have not shown that $(0,0)$ is the zero vector. For any $(x,y)\in V$, $(x,y)+(0,0)=(0,0)+(x,y)=(x,y)$. Therefore, $(0,0)$ is the identity element and hence, the zero vector.

