I'm used to thinking of $\mathbb{R}^1$ as (basically) the $x$-axis in one dimension. Similarly, I think of $\mathbb{R}^2$ as the $xy$-plane in two dimensions (i.e. in contrast to the plane $z = 0$ in Cartesian space / $\mathbb{R}^3$).
This aligns with the fact that $\mathbb{R}^1 \not\subset \mathbb{R}^2$ and $\mathbb{R}^1, \, \mathbb{R}^2 \not\subset \mathbb{R}^3$; however, I'm left wondering: is there a conventional way of referring to the subspaces of $\mathbb{R}^3$ that are isomorphic to $\mathbb{R}^1$ and $\mathbb{R}^2$?
Thank you in advance for your time and help!
Edit:
Basically I'm wondering if there is any convention for a shorthand to refer to $\text{Span}\left\{ \, \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \, \right\}$ and $\text{Span}\left\{ \, \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix},\, \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} \, \right\}$ (as well as analogous subspaces in higher dimensions)?
An example of a situation I would like to use this notation in: a succinct way of asking if a vector in $\mathbb{R}^3$ is a basis for $\text{Span}\left\{ \,\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \, \right\}$.
Obviously asking "is $\text{Span}\left\{ \, \vec{v}_1 \, \right\} = \mathbb{R}^1$" would be incorrect, and "is $\text{Span}\left\{ \, \vec{v}_1 \, \right\}$ isomorphic to $\mathbb{R}^1$" isn't what I'm looking for either.
Your problem is thinking about "the" subspace of $\mathbb{R}^3$ that is (isomorphic to) $\mathbb{R}$. There are infinitely many. You can single out the three coordinate axes as special since you are representing vectors as ordered triples, but the $x$-axis is no more special than the other two axes and does not have another common name. Even calling the $x$-axis relies on the convention that the axes are labelled $x$, $y$ and $z$. In higher dimensions that won't work.