My linear algebra class recently covered isomorphisms between vector spaces. Much emphasis was placed on the fact that $\mathbb{R}^m$ is isomorphic to particular subsets of $\mathbb{R}^n$ for positive $m < n$. For example, $\mathbb{R}^2$ is isomorphic to every plane in $\mathbb{R}^3$ passing through the origin.
My question is whether $\mathbb{R}$ is isomorphic to $\mathbb{R}^n$ for all $n > 1$. It follows from the fact that $\lvert \mathbb{R} \rvert = \lvert \mathbb{R}^n \rvert$ that there exist bijections $\phi : \mathbb{R} \rightarrow \mathbb{R}^n$ for all $n > 1$, but showing that these sets are isomorphic as vector spaces would require showing that such $\phi$ are linear. The standard constructions don't appear to give linear maps, but that doesn't preclude the existence of some linear mapping.
I'm rather surprised that my book didn't answer this question, because it seems like a natural question to ask. That, or the answer is obvious and I'm overlooking some important fact.
Best Answer
I have absolutely no doubt that your book covers this point. Two vector spaces are isomorphic if and only if they have the same dimension. So $\mathbb{R}$ (dimension $1$) cannot be isomorphic to any $\mathbb{R}^n$ (dimension $n$) if $n>1$.