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

Question

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 ?

then as @Fred says you are wrong in your reasoning, though you have chosen the right answer — ancient mathematician, Sep 09 '19 at 09:21
@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.) — Arthur, Sep 09 '19 at 09:22
As clarified over here, the problem seems to lie at $\Bbb R\oplus\Bbb R\cong 2\Bbb R$. — M. Winter, Sep 10 '19 at 09:41

score 3 · Accepted Answer · answered Sep 09 '19 at 09:20

3

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

answered Sep 09 '19 at 09:20

Fred