Is $\mathbb{R}$ a subspace of $\mathbb{R}^2$? I think I've been confusing myself about the language of subspaces and so on.  This is a rather basic question, so please bare with me.  I'm wondering why we do not (or perhaps "we" do, and I just don't know about it) say that $\mathbb{ R } $ is a subspace of $\mathbb{ R }^2 $.  It's elementary to prove that the set
$$ S:= \left\{ c \cdot \mathbf{x} \mid c \in \mathbb{ R }, \mathbf{x} \in \mathbb{ R }^2  \right\}$$
is a vector subspace of $\mathbb{ R } ^2$.  What is confusing me is that there seems to be an isomorphism between the set $S$ and $\mathbb{ R } $:
\begin{align*}
\varphi: S &\rightarrow \mathbb{ R }   \\
 c \cdot \mathbf{x} &\mapsto c \\
\end{align*}
If this is indeed true, as I believe it is having checked that $\varphi$ gives an isomorphism, wouldn't we say that $\mathbb{ R } $ is a subspace of $\mathbb{ R } ^2$?
Any help sorting out this (language) problem will be greatly appreciated!  

This is indeed an important question. No doubt that $\mathbb{R}$ is isomorphic to many subspaces of $\mathbb{R}^2$, or in other words, $\mathbb{R}$ can be embedded in many ways in $\mathbb{R}^2$. The thing is that there isn't any specific subspace in $\mathbb{R}^2$ which is the best one to represent $\mathbb{R}$.
If you want to treat $\mathbb{R}$ as a subspace, you need to specify the embedding $\mathbb{R}\hookrightarrow\mathbb{R}^2$ you refer to. 
I would say that as long as we don't choose the embedding, $\mathbb{R}$ is not a subspace of $\mathbb{R}^2$.
