Zero Vector for Different Vector Spaces

Question

The zero vector of subspaces is the same for the entire space; is this true if you have two (or more) separate vector spaces? Intuition says yes, since I imagine the zero vector to be at the origin and the origin is fixed, so the zero vector would be the same for all possible vector spaces. I've just never seen it stated this way explicitly, can anyone confirm the logic please?

Answer 1

Geometrically, the zero vector has the same meaning in all vector spaces; it lies at the origin (whatever that means when you're talking about an infinite dimensional space...). However, the zero vector is not the same for different vector spaces. A vector space consists of a set endowed with some operations and a field underlying it; if the sets are different, the vectors are completely different.

Specific examples: In $\mathbb{R}^2$, the zero vector is the vector $[0, 0]$. In $\mathbb{R}^3$, it is $[0, 0, 0]$. In the space of continuous functions from $\mathbb{C} \to \mathbb{C}$, it is the zero function on $\mathbb{C}$.