Basis for a vector space consisting of the set of linear combinations of these functions S is the vector space consisting of the set of all linear combinations of the functions $f_1(x)=e^x$, $f_2(x)=e^{-x}$, $f_3=sinh(x)$. Find a basis for S and find the dim[S]. 
First, I let 
$c_1f_1$+$c_2f_2$+$c_3f_3$=$f$
Then, since $sinh(x) =\frac{e^x-e^{-x}}{2}$, 
$f_3=\frac{f_1-f_2}{2}$
Substituting back in, I can get 
$c_1f_1$+$c_2f_2$+$c_3\frac{f_1-f_2}{2}$ =$f$
$(c_1+\frac{1}{2}c_3)f_1$+$(c_2-\frac{1}{2}c_3)f_2$ =$f$
Am I on to something? 

$f_1$ and $f_2$ are linearly independent (check using the definition), while $f_3$ is a linear combination of the first two. So the basis (i.e. linearly independent set which spans the space) is just $\{f_1,f_2\}$. Since this set contains two elements, the space is two dimensional. 
