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?

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.
• Be extra careful. You have not yet proved that $(0,0)$ is the zero vector. – edm Oct 30 '18 at 4:15