Eigenfunctions for distinct eigenvalues are never linearly dependent

Let $\lambda_1\neq\lambda_2$ be the characteristic numbers and $\phi_1,\phi_2$ be the corresponding eigenfunctions for the homogeneous Fredholm integral equation with separable kernel $u(t)=\lambda\int_0^1k(t,s)u(s)ds$. Then which of the following is not true?