Answer
See below
Work Step by Step
Given $V=C^\infty(I)$
and $S$ be the subspace of $V$
a) For any vector $r(x) \in span \{f(x),g(x)\}$ we have
$r(x)=af(x)+bf(x)\\
=a\frac{e^x+e^{-x}}{2}+b\frac{e^x-e^{-x}}{2}\\
=\frac{ae^x+ae^{-x}}{2}+\frac{be^x-be^{-x}}{2}\\
=\frac{a+b}{2}e^x+\frac{a-b}{2}e^{-x}$
b) From exercise a) we can notice that any vector $r(x) \in$ span $\{f(x),g(x)\}$ can be written as a linear combination of $h(x)=e^{-x}$ and $j(x)=e^{-x}$
Hence, $S$ is also spanned by $\{h(x), j(x)\}$