Is a vector in $\mathbb{R}^n$ always a column vector? I suppose the essence is in the title. If I write $\mathbf{v} \in \mathbb{R}^n$, does that clearly define $\mathbf{v}$ to be a column vector, or is it more like saying '$n$ real numbers in some arbitrary arrangement'? As a follow up, if I need to specify that $\mathbf{v}$ is a column vector—not a row vector or anything else—would $\mathbf{v} \in \mathbb{R}^{n \times 1}$ be the proper way?
 A: Definitely NOT! Column vectors are just a point of view to consider the vectors in $\mathbb{R}^n$ which mathematicians usually assumed. However, it is not the only way! For example, if $n=m\times m$, then you can express your vectors as $m\times m$ matrices. You can also write your vectors as row vectors, then the linear transformations will be the transport while it is not in the situation of column vectors. That is, we usually write $$Ax$$ for column vector $x$ and linear transformation $A$, but you need to write $$x^TA^T$$ if you want to express it into row vectors, here $x$ is the original column vector.
A: Columns and tuples (not rows) are usually used to represent "column vectors" in $\mathbb R^n$, so yes, if you write $v\in\mathbb R^n$, it should be sufficient.
In linear algebra, a "row vector" (which is actually a linear functional) is usually represented as $v^T$ for some $v\in\mathbb R^n$.
A: Formally an element of $\mathbb{R}^{n}$ is an element of the direct product of $n$ copies of the field $\mathbb{R}$, that is an $n$-tuple $(x_{1}, \dots, x_{n})$ of real numbers, so is not a priori identified with either a column or a row vector. A column vector can be viewed as the matrix representing a linear map from $\mathbb{R}$ to $\mathbb{R}^{n}$ with respect to some choices of bases in these spaces, and a row vector can be viewed as the matrix representing a linear map from $\mathbb{R}^{n}$ to $\mathbb{R}$ with respect to some choices of bases in these spaces. The vector spaces of $n$-tuples of real numbers, $n$-fold column vectors, and $n$-fold row vectors are not canonically identified, although they are all isomorphic, so it is probably best to use notations that distinguish them, for example $\mathbb{R}^{n}$, $\mathbb{R}^{n\times 1}$, and $\mathbb{R}^{1 \times n}$ (whether these are good notations is a different matter), although such careful distinctions can confuse students at first. 
A: You can write as row vector or as a column vector. You no need to specify anything like that. But when we do linear algebra it is always convenient to see vector as column vector. 
