Exercise 4

Prove that each set $\{\hskip 1.00006ptf,g\hskip 1.00006pt\}$ is linearly independent in the vector space of all functions from $\mathbb{R}^{+}$ to $\mathbb{R}$.

• $f(x)=x$ and $g(x)=1/x$

• $f(x)=\cos(x)$ and $g(x)=\sin(x)$

• $f(x)=e^{x}$ and $g(x)=\ln(x)$