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Step 1

Note that {\mathbb  {R}} is a vector space over the field {\mathbb  {Q}} of rational numbers, with ordinary real multiplication as the scalar multiplication. Let H be the basis of that space (a so-called Hamel basis). The dimension over {\mathbb  {Q}} of the defined space is |H|=c=2^{{\aleph _{0}}}.

Step 2

Note that {\mathbb  {R}}^{2} is also a vector space over the field {\mathbb  {Q}}; it has a basis (H\times \{0\})\cup (\{0\}\times H) so its dimension is |(H\times \{0\})\cup (\{0\}\times H)|=2|H|=|H| because H is infinite, so this dimension is c too.

Step 3

Both {\mathbb  {R}} and {\mathbb  {R}}^{2} are vector spaces over {\mathbb  {Q}} of the same dimension; therefore they are isomorphic (as vector spaces). However, every isomorphism of linear spaces is additive and for that reason it is also an isomorphism of their additive groups. We have proven that ({\mathbb  {R}},+)\simeq ({\mathbb  {R}}^{2},+).

Step 4

The isomorphism between ({\mathbb  {R}}^{n},+) and ({\mathbb  {R}}^{m},+) is a trivial consequence of the isomorphism proven above.

Abstract Algebra

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