TITLE: Zero vector of a vector space
QUESTION [21 upvotes]: I know that every vector space needs to contain a zero vector. But all the vector spaces I've seen have the zero vector actually being zero (e.g. $\mathbf{0}=\langle0,0,\ldots,0\rangle$). Can't the "zero vector" not involve zero, as long as it acts as the additive identity? If that's the case then are there any graphical representations of a vector space that does not contain the origin?

REPLY [2 votes]: One more comment- in the specific vector space, $\mathbb R^n$, the zero vector is the $n$-tuple $(0, 0, \ldots, 0)$.  Any $n$-dimensional vector space is "isomorphic" to $R^n$, with, given a basis $e_1, e_2, \ldots , e_n$, the isomorphism that maps $e_n$ to $(0, 0, \ldots, 1, \ldots, 0)$ where the "$1$" is in the $n$th place so we can always express vectors in the $(a, b, \ldots, z)$ notation.
However there exist infinite dimensional vector spaces for which we cannot use that notation.