A distance preserving isomorphism from $R^{3}$ to $R^{2}$

Question

Does there exist a distance preserving isomorphism $I: \mathbb{R}^{3} \rightarrow\mathbb{R}^{2}$, where both $\mathbb{R}^{3}$ and $\ \mathbb{R}^{2}$ have their respective euclidean metrics? More generally, is there such an isomorphism from $\mathbb{R}^{n}$ to $\mathbb{R}^{2}$?

Answer 1

As has already been said, you cannot have an isomorphism between vector spaces of different dimensions. However, suppose instead you meant 'mapping' between $\mathbb{R}^n$ and $\mathbb{R}^2$. If your vector in $\mathbb{R}^n$ is

$$(x_1,x_2,\cdots, x_n)\in \mathbb{R}^n$$

then one such mapping is

$$X=\sqrt{x_1^2+\frac{1}{2}\left(x_3^2+x_4^2+\cdots+x_n^2\right)}$$

$$Y=\sqrt{x_2^2+\frac{1}{2}\left(x_3^2+x_4^2+\cdots+x_n^2\right)}.$$

Of course, there are infinite mappings which preserve the length of the original vector, some of which are simply in $\mathbb{R}$

$$X=\sqrt{x_1^2+x_2^2+\cdots+x_n^2}$$

Without knowing more about your specific problem, it's hard to go further than this.