Is $\mathbb{R^n}$ the same as $\mathbb{R^{n \times 1}}$, or is there a fundamental difference? Is it that $\mathbb{R^{n \times 1}}$ implies we are dealing with a something else, despite an element of $\mathbb{R^{n \times 1}}$ looking exactly the same as $\mathbb{R^n}$. And if there is no difference:
Why would one even take the time to write $\mathbb{R^{n \times 1}}$ instead of $\mathbb{R^n}$?
 A: You might write $\mathbb{R}^{n\times 1}$ 'column-vectors' instead of $\mathbb{R}^n$ to explicitly differentiate $\mathbb{R}^{n\times 1}$ from its dual, which is representable by $\mathbb{R}^{1\times n},$ 'row-vectors.'
One reason to do this is to make when you are dealing with adjoints explicit; In some technical sense, $a\in \mathbb{R}^n$ and $a^\dagger$ are not the same type of object: $a^\dagger$ is a bounded linear function $\mathbb{R}^n\to \mathbb{R}$, whereas $a$ is just something that $a^\dagger$ might act on.
A: As a set or a topological space, elements of $\mathbb{R}^n$ are usually written as $n$-tuples $(a_1, \dotsc, a_n)$. As a vector space, elements of $\mathbb{R}^n$ are written either as column vectors $$\begin{bmatrix}a_1\\ \vdots\\a_n\end{bmatrix} \tag{*}$$
or also as $n$-tuples $(a_1, \dotsc, a_n)$ and I've seen $\langle a_1, \dotsc, a_n \rangle$ as well. 
The space of $m$-by-$n$ matrices ($m$ rows and $n$ columns) with coefficients in $\mathbb{R}$ is sometimes denoted $\mathbb{R}^{m \times n}$. With this convention, a typical element of $\mathbb{R}^{n \times 1}$ would be an $n$-by-$1$ column matrix, notationally indistinguishable from the vector in (*).
