X is to vector like matrix is to linear operator? In linear algebra texts there is usually a clear distinction between linear operators and matrices.


*

*A linear operator is a map between two spaces that fulfills a set of conditions.

*A matrix is a 2D array of numbers (elements of the underlying field) that can be used to represent a linear operator with respect to bases of the two spaces.
Slightly complicating but not muddling this distinction is the fact that together with appropriate operations, a matrix can also be seen as a linear operator itself.
What I'm missing is a corresponding distinction for vectors. The term "vector" is interchangeably used for elements of a space, and for their representation by 1D arrays with respect to a basis.
Is there terminology to distinguish between these two meanings of "vector", like there is for "linear operator" vs "matrix"?
 A: What you are looking for is the (coordinate vector) of a vector with respect to  a given basis. 
As you know the same vector has different coordinates in different bases and usually the standard basis is used. 
A: Here is one way to think about this: if we fix a basis $\mathcal{B}=\{v_1,\ldots, v_n\}$ for $V$, there is an associated dual basis $\mathcal{B}^*=\{\phi^1,\ldots, \phi^n\}$ for $V^*$. Just as a matrix $A\in \mathcal{M}_n(\mathbb{R})$ represents a linear map $T:V\to V$ viewed in $\mathcal{B}$ coordinates, we can view a "column vector" (i.e. $n\times 1$ array of numbers) as representing an element of $V$. 
In the case of $A$ representing $T$, we know that the columns of $A$ codify what $T$ does to each basis vector $v_i$, i.e. $A=(a^i_j)$ where 
$$ v_j=\sum_{i=1}^na^i_jv_i.$$
Fix $v\in V$: we can provide an analogous interpretation. $v$ defines a linear map $V^*\to \mathbb{R}$ (or the ground field of your choosing) by $\phi\mapsto \phi(v).$ Then, viewed as a linear map, $v$  has a "matrix representation" $[v]_{\mathcal{B}}$ consisting of what $v$ does to the elements of the basis: namely 
$$ [v]_{\mathcal{B}}=
\begin{bmatrix}
\phi^1(v)\\
\vdots\\
\phi^n(v)
\end{bmatrix}.$$
You may object slightly by saying that this array should be $1\times n$ because it is mapping from an $n-$dimensional space to a $1-$dimensional space. However, the privilege of being an $n\times 1$ array is reserved for elements of $V^*$ so that we can define a pairing $\langle \:,\:\rangle:V^*\times V\to \mathbb{R}$ by $$\langle \phi,v\rangle=\underbrace{[\phi]_{\mathcal{B}^*}\times[v]_{\mathcal{B}}}_{\text{as matrices}}\in \mathbb{R}.$$
A: Typically, vectors are associated with coordinates in a finite-dimensional vector space (e.g. $\mathbb{R}^n$).  An important distinction is that linear spaces are much more general than finite-dimensional vector spaces.  So, you can view a matrix as a linear operator that acts in a finite dimensional vector space.  However, linear operators are much more general in that they can be associated with linear spaces which may be infinite-dimensional.  So, matrices are a small class of linear operators.  By analogy, a vector is just one type of linear space element.  For example, you might have a linear space where each 'element' is a function rather than a vector.  So, you might say that matrices are to linear operators what vectors are to elements of general linear spaces.
