  This is the Part I paper of our series of full papers on a gauge-invariant
{\it linear} perturbation theory on the Schwarzschild background spacetime
which was briefly reported in our short papers [K.~Nakamura, Class. Quantum
Grav. {\bf 38} (2021), 145010; K.~Nakamura, Letters in High Energy Physics {\bf
2021} (2021), 215.]. We first review our general framework of the
gauge-invariant perturbation theory, which can be easily extended to the {\it
higher-order} perturbation theory. When we apply this general framework to
perturbations on the Schwarzschild background spacetime, a gauge-invariant
treatments of $l=0,1$ mode perturbations are required. On the other hand, in
the current consensus on the perturbations of the Schwarzschild spacetime,
gauge-invariant treatments for $l=0,1$ modes are difficult if we keep the
reconstruction of the original metric perturbations in our mind. Based on this
situation, we propose a strategy of a gauge-invariant treatments of $l=0,1$
mode perturbations through the decomposition of the metric perturbations by
singular harmonic functions at once and the regularization of this singularity
through the imposition of the boundary conditions to the Einstein equations.
Following this proposal, we derive the linearized Einstein equations for any
modes of $l\geq 0$ in a gauge-invariant manner. We discuss the solutions to the
odd-mode perturbation equations in the linearized Einstein equations and show
that these perturbations include the Kerr parameter perturbation in these
odd-mode perturbation, which is physically reasonable.
