Does $\mathbb{R}^n$ have a real vector space structure with dimension other than $n$?

Can we define a vector space structure on $\mathbb {R}^n$ other than usual scalar multiplication and usual addition such that the dimension of $\mathbb {R}^n$ over $\mathbb {R}$ is not $n$ but some $m$ not equal to $n$?

Solution 1:

Yes we can. Let $m, n \geq 1$, $m \neq n$.

As $\mathbb{R}^m$ and $\mathbb{R}^n$ have the same cardinality, there is a bijection $\varphi : \mathbb{R}^m \to \mathbb{R}^n$. Now we can define an alternative vector space structure on $\mathbb{R}^n$ as follows:

• for $a \in \mathbb{R}$ and $v \in \mathbb{R}^n$, $a\cdot v := \varphi(a\varphi^{-1}(v))$,
• for $v, w \in \mathbb{R}^n$, $v + w := \varphi(\varphi^{-1}(v) + \varphi^{-1}(w))$.

You can verify that all of the axioms are satisfied (the zero vector is $\varphi(0)$, and the additive inverse of $v$ is $\varphi(-\varphi^{-1}(v))$).

The dimension of this vector space is not $n$ but rather $m$. An explicit basis is given by $\{\varphi(e_1), \dots, \varphi(e_m)\}$.

This is an example of a transport of structure.

Solution 2:

Consider $\mathbb{R}^n$ as a vector space over $\mathbb{Q}$. The dimension is $\infty$.