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If we look at $\mathbb{C}$ as a vector space over $\mathbb{R}$ it's dimension will be $2$, because $\mathbb{C} = span\{1,i\}$.
A question I thought of is what would be the dimension of $\mathbb{R}$ as a vector space over $\mathbb{Q}$?
I feel like the answer should be infinity, because if the dimension was finite, say $n$, then for every $m := n+1$ real numbers $x_1,...x_m$ there was a linear combination with rational coefficients that gives $0$: $\frac{a_1}{b_1}x_1+...+\frac{a_m}{b_m}x_m = 0$. Multiplying by $lcm(b_1,...,b_m)$ we get that for every $m$ real numbers there is a linear combination with natural coefficients that gives $0$. That feels false, how do you prove it?

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marked as duplicate by José Carlos Santos linear-algebra May 8 at 16:52

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