Yes/No :$\mathbb{R}$ is isomorphic to $\mathbb{R}\oplus \mathbb{R}$ as vector spaces over $\mathbb{Q}$

My attempt : yes

i think $\mathbb{R}\cong \mathbb{R}\oplus \mathbb{R} \cong2 \mathbb{R}$ both have same dimension that is dim$( \mathbb{R} )= 1$

Is its true ?

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    $\begingroup$ You say $\dim(\mathbb{R})=1$. Dimension over what? $\endgroup$ Sep 9 '19 at 9:19
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    $\begingroup$ @jasmine If the dimension of $\Bbb R$ as a vector field over $\Bbb Q$ is $1$, then all elements are in the $\Bbb Q$-span of a single non-zero real number, say $1\in \Bbb R$. Can you express $\sqrt 2$ as a linear combination of $1$ with rational coefficients? (It seems strange to write "linear combination" with only one vector, but still.) $\endgroup$
    – Arthur
    Sep 9 '19 at 9:22
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    $\begingroup$ math.stackexchange.com/questions/26781 $\endgroup$ Sep 9 '19 at 9:24
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    $\begingroup$ thanks u @ Fred $\endgroup$
    – Fred
    Sep 9 '19 at 9:41
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    $\begingroup$ As clarified over here, the problem seems to lie at $\Bbb R\oplus\Bbb R\cong 2\Bbb R$. $\endgroup$
    – M. Winter
    Sep 10 '19 at 9:41

$ \mathbb R$ as a vector space over $ \mathbb Q$ is not finite-dimensional !


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