Question regarding notation involving vector spaces. Let V be the set of all ordered pairs of real numbers, and consider the following addition and scalar multiplication operations on $u+v=(u_1+v_1+1,u_2+v_2+1),$ $ku=(ku_1,ku_2)$.
Show that $(0,0)\neq0$.
I'm confused by their notation. Can anyone explain?

Let $\mathbb{V}$ be the vector space on V with the above definitions of addition and scalar multiplication. In the statement you are to show, the zeroes on the left side are real numbers making up the components of a vector in $\mathbb{V}$. The zero on the right side is not a real number but a vector in $\mathbb{V}$, namely the zero vector in $\mathbb{V}$. You are to show that the vector $(0,0)$ is not equal to the zero vector in $\mathbb{V}$.
