Dimension of a vector space below I have to prove that the dimension of the vector space of real numbers over Q (rational numbers) is infinity. How can I prove? I have no idea.

Remember that the dimension of a vector space is finite means that there is a finite basis. 
In your case, the "vectors" are real numbers and the "scalars" are rational numbers. So the dimension being finite means there is a finite set $r_1,\ldots,r_n$ of real numbers such that every real number $r$ can be written in the form $r=q_1r_1+q_2r_2+\cdots q_nr_n,$ for some rational numbers $q_1,\ldots,q_n$. Can you see the contradiction here by thinking about countability vs uncountability?
