I think you mean that you are given a pair of vectors $\{(4,1),(-7,-8)\}$ and asking whether or not they form a basis for $\mathbb{R}^2$. If you are allowed to use the fact that the dimension of a vector space is well-defined, all you need to prove is that the vectors are linearly independent or that they span the space (as either of these, the fact that dim$(\mathbb{R}) = 2$), implies the other); otherwise you must prove both.
To prove that the vectors are linearly independent, try to solve the equation $a(4,1)+b(-7,-8)=(0,0)$ and show that no solution exists.
To prove that the vectors span the space, show that $(1,0)$ and $(0,1)$ can be written as a linear combination of the vectors you are given, thus any vector in $\mathbb{R}^2$ can be written as such.
Hope that helps.