True or false: and are isomorphic over

TIFR GS 2019, Part B Problem 10

Linear algebra

Easy

Linear algebra, Insel, Frieberg

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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 \(\Bbb Q\).

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and \(dim_{\Bbb Q}(\Bbb R \oplus \Bbb R)=\mathfrak N_0\). ### Hence, \(\Bbb R\) and \(\Bbb R \oplus \Bbb R\) are isomorphic over \(\Bbb Q\).

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