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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 $n$?

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    $\begingroup$ Dimension doesn't depends on scalar multiplication or smth. else $\endgroup$ Aug 19, 2015 at 16:36

2 Answers 2

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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.

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  • $\begingroup$ but in the second point, v,w are taken from ${R}^m$, this should be from ${R}^n$ $\endgroup$
    – Puneet
    Aug 19, 2015 at 16:57
  • $\begingroup$ Can you suggest some good books of linear algebra, where I can find these kind of interesting questions $\endgroup$
    – Puneet
    Aug 19, 2015 at 17:00
  • $\begingroup$ @Puneet: Yes, I did mean $\mathbb{R}^n$, thanks. I don't know of a book that discusses this, but I have added a link which describes this type of construction. $\endgroup$ Aug 19, 2015 at 17:14
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Consider $\mathbb{R}^n$ as a vector space over $\mathbb{Q}$. The dimension is $\infty$.

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  • $\begingroup$ if m is finite, then ? $\endgroup$
    – Puneet
    Aug 19, 2015 at 16:47
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    $\begingroup$ The question was asking for a vector space structure over $\mathbb{R}$. $\endgroup$ Aug 19, 2015 at 17:07

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