Is a coordinate vector an element of $M_{n\times 1}$ or $F^n$? Let $V$ be a finite-dimensional vector space with an ordered base $\beta$ over a field $F$.
Let $x\in V$ an $[x]_{\beta}$ be a coordinate vector of $x$ relative to $\beta$.
Is this an element of $M_{n\times 1}(F)$ or $F^n$?
Since we usually treat them as a matrix, i think it should be defined as a matrix so that $[x]_{\beta} \in M_{n\times 1}(F)$. However, the text i'm studying states that it is an element of $F^n$. (Note that $F^n\neq M_{n\times 1}(F)=F^{n\times 1}$)
The reason why I'm distinguishing these two even though they have the same meaning, is to formally define a matrix multiplication. For example, an element in $F^n$ is not only sometimes treated as a column vector, but also treated as a row vector. If these two are not distinguished then multiplication cannot be well defined.
Am i correct? Or if i am wrong or if what i'm worrying is not a matter, please explain me why. Thank you in advance.
 A: You are absolutely correct.
However, -- and if you read some specific text, then it's probably written somewhere explicitly -- it is usual to identify $F^n$ with the culomn vectors, i.e. $n\times 1$ matrices. As you say, $F^n\ne M_{n\times 1}(F)$, but they are isomorphic. It is also usual that an $n\times m$ matrix $A$ represents a linear map $F^m\to F^n$, namely
$$v\mapsto A\cdot v$$
where the product is given by matrix multiplication. Here $v$ was considered as a culomn vector (this is the default, if composition of (linear) functions is written to the left: $f\circ g=x\mapsto f(g(x))$.)
A: The purpose of a coordinate vector is so that you can turn problems into matrix algebra; $M_{n \times 1}(F)$ is the best choice for what a coordinate vector should be if you're into such fine detail.
But it really is a fine detail. Part of the power of linear and multi-linear algebra is how smoothly you can shift between many different interpretations of an object. e.g. an element of $M_{n \times n}(F)$ can be thought of as a linear transformation on $M_{n \times 1}(F)$, a linear transformation on $M_{1 \times n}(F)$, a linear transformation on any $M_{n \times m}(F)$, a linear transformation on $V = F^n$, a linear transformation on $V^*$, an element of $(V^*)^n$, an element of $(V^n)^*$, an element of $M_{2 \times 2}(M_{n/2 \times n/2}(F))$, ..., a full rank matrix in $M_{m \times n}(F)$ with $m < n$ can be viewed as a subspace of $F^n$, ....
