Formalisation of equivalence of "vectors" and "points" In all the linear algebra courses I've taken, the notion of a "geometric vector" has not been rigorously defined. All that has been explained is that we can think of them as algebraically equivalent to points, but geometrically they can be "moved" around without changing. How do you define these things formally? Below is my attempt; is it adequate, and are there better definitions out there?
A geometric vector in $\mathbb{R}^N$ is an element of $\mathbb{R}^{2N}/\text{~}$, where ~ is defined as follows: Writing a general element of $\mathbb{R}^{2N}$ as (tail, tip) or $(\bf{a}, \bf{b})$ for $\bf{a}, \bf{b}$ $\in \mathbb{R}^N$, we have
$$(\bf{a}, \bf{b}) \; \text{~} \; (\bf{c}, \bf{d}) \leftrightarrow \bf{d} - \bf{b} = \bf{c} - \bf{a}.$$
The vector space isomorphism between this quotient (endowed with fairly obvious operations) and $\mathbb{R}^N$ is not hard to show. Also, (tail$_1$, tip$_1$) ~ (tail$_2$, tip$_2$) exactly when you can "move" the corresponding vectors to each other, so ~ captures the concept of "moving".
Thanks for the help.
 A: The key is the notion of affine space (as opposed to vector space). 
A vector space is a set $E$ together with an additive law $+$ and scalar multiplication (satisfying the usual axioms). Naturally $\mathbb{R}^n$ is endowed with a vector space structure. 
An affine space $A$ directed by a vector space $E$ is a set $A$ together with a "faithful and transitive group action of $E$", i.e. a group morphism 
$t : E \rightarrow B(A)$ (where $B(A)$ is the set of bijections of $A$) such that :


*

*for all $a,b \in A$, there is a $x \in E$ with $t(x)a=b$  (transitive)

*for all $x \in E$, if $x \neq 0$ then $t(x)$ is not the identity map (faithful)


What this formal definition represents is $t(x)$ is the translation of vector $x$. The elements of $A$ are points, which are moved by vectors through the action of $t$. Note that if $E$ is a vector space then it has a natural affine structure. This is the most formal definition of affine spaces/vector spaces, which has at least the advantage of clearly separating the different concepts (points and vectors). Hope that helps !
A: In each mathematical system you will find basic concepts that won't/can't be defined, their interrelationships are as described by the relevant axioms. Think "elements" and "operations" in a group. Yes, you can give examples ("look, the integers $\mathbb{Z}$ with $+$ form a group") or give some other analogy to give a mental picture ("a vector can be considered a point in space, and ..."), but what you really have to understand is the abstract aspect.
[Somebody said about mathematics that "abstract algebra is the subject that separates adults from children"; meaning that this subject is hard, and a thorough understanding is the basis for further advances.]
