The given set of functions is $$[ \sinh(x),\space\cosh(x), e^x]$$ $\begin{bmatrix} \sinh(x) \space \cosh(x) \space e^{-x}\\ \cosh(x) \space \space \sinh(x) \space -e^{-x}\\ \sinh(x) \space \cosh(x) \space e^{-x} \end{bmatrix}$
Using the matrix above I multiplied through, taking the Wronskian of the given functions and it's given derivatives. After further factoring and addition I calculated the equation to be $$e^{-x}(\cosh(2x) +\sinh(2x))=0 $$ Then using hyperbolic identities to rewrite the trig. terms I got $$ e^{-x} \left(\frac{e^{2x}+e^{-2x}+e^{2x}-e^{-2x}}{2} \right)=0$$ $$e^{-x}(e^{2x})=0$$ $$e^{-x} \ne0$$ There for the set of functions is linearly independent. Are my calculations and verdict correct?
