What is a vector and what is a change of basis I've just started my first course in linear algebra and I am confused.
If I have a based free module $M$ with basis $\{e_1,\dots,e_n\}$, then I can write any vector as $\sum_{i=1}^n a_ie_i$.
Say I consider $\Bbb C^n$, then any two $n$-dimensional vector spaces $V$ and $W$ with bases $\{v_1,\dots,v_n\}$ and $\{w_1,\dots,w_n\}$. Then if I want to take a change of basis from the basis of $V$ to the basis of $W$, then I can do this with any isomorphism $V\to W$, is that right? So if I solely want to send the $v_i$ to $w_j$ for some $i$ and $j$, then I have $n!$ ways of doing this. But I could also send $v_1$ to $w_1+w_2$ and $v_2$ to $w_1-w_2$, would that still be a change of basis?
What is a change of basis? Also an element of a vector space is a vector, but what is a vector fundamentally? The vector can take different forms depending on the choice of basis? I want to say something like, a vector $x\in \Bbb C^n$ is a representative for a class of vectors, one for each choice of basis?
 A: Consider the vector space of distances: $\Bbb R \vec m$, where $\vec m$ stands for meters. The vectors of this space are called distances; for instance, $1\vec m$, (read "one meter"), is a vector of that space. All of a sudden you get interested in the size of ants and meters are no longer convenient. You then decide to measure distances in centimeters. You write $\vec{cm} = 0.01 \vec{m}$. $\{\vec m\}$ is your old basis and $\{\vec{cm}\}$ is the new basis. Say you have some distance $d$. You can write $d$ in meters as $d=d_m \vec m$ or in centimeters as $d = d_{cm} \vec{cm}$; of course $d_m=d_{cm}/100$. This is the same vector written in two different bases; this is a change of basis.
A change of basis refers to changing how vectors are expanded in one and the same space. In the example, we have $\Bbb R \vec m = \Bbb R \vec{cm}$. This is a "hard" equality not just an equality up to an isomorphism.
A vector is fundamentally ... an element of a vector space. But it does not hurt to think of it as an arrow of some length pointing in some direction (to begin with at least).
Your intuition that $x\in \Bbb C^n$ is a representative of a class of vectors is not quite right. It is rather that $\Bbb C^n$ is a representative of a class of vector spaces. Further, if you take one of these spaces, call it $E$, there is one isomorphism between $\Bbb C^n$ and $E$, for each choice of basis of $E$. In the example above, there is one isomorphism between $\Bbb R$ and $\Bbb R\vec m$ for each choice of length units: kilometers, feet, lightyears, ... But $\Bbb R$ is not really equal to $\Bbb R\vec m$. Same as how $\Bbb R \vec{m}$ is not equal to $\Bbb R \vec{kg}$ even though they are isomorphic.
