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# Understand the problem

True or false: $\Bbb R \oplus \Bbb R$ and $\Bbb R$ are isomorphic over $\Bbb Q$
##### Source of the problem
TIFR GS 2019, Part B Problem 10
Linear algebra
Easy
##### Suggested Book
Linear algebra, Insel, Frieberg

Do you really need a hint? Try it first!

Use the fact that even in infinite dimensional vector space, equal dimensions give us isomorphic vector space.
$$dim_{\Bbb Q}(\Bbb R )=\mathfrak N_0$$ and $$dim_{\Bbb Q}(\Bbb R \oplus \Bbb R)=\mathfrak N_0$$.

### Hence, $$\Bbb R$$ and $$\Bbb R \oplus \Bbb R$$ are isomorphic over $$\Bbb Q$$.

Do you really need a hint? Try it first!

Use the fact that even in infinite dimensional vector space, equal dimensions give us isomorphic vector space.
$dim_{\Bbb Q}(\Bbb R)=\mathfrak N_{0}$ and $$dim_{\Bbb Q}(\Bbb R \oplus \Bbb R)=\mathfrak N_0$$.

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