# what is the dimension of $\mathbb{R}$ at a vector space over there field $\mathbb{Q}$? [duplicate]

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?

## marked as duplicate by José Carlos Santos linear-algebra StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); May 8 at 16:52

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