True or False: Every finite dimensional vector space can made into an inner product space with the same dimension. Every finite dimensional vector space can made into an inner product space with the
same dimension.

I think it depends on what field you are using for your vector space.  If it is $\mathbf{R}$ or $\mathbb{C}$, the answer is definitely "yes" (see the comments, which are correct).  I am pretty sure it is "yes" if your field is a subfield of $\mathbb{C}$ that is closed (the word "stable" also seems to be standard) under complex conjugation, such as the algebraic numbers. Otherwise, e.g. if your field is $\mathbf{F}_2$, I don't know.  I consulted Wikipedia's article on inner product spaces and they only dealt with the case where the field was the reals or the complex numbers.
EDIT: Marc's answer is better than mine.  See his comment regaring subfields of $\mathbf{C}$.  Some such subfields are not stable under complex conjugation and cannot be used as a field for an inner product space. I am pretty sure that if $\mathbb{K}$ is any ordered field (which may or may not be a subfield of $\mathbb{R}$) you can use it as the field for an inner product space. 
