  We demonstrate the fact that linearity is a meaningful symmetry in the sense
of Lie and Noether. The role played by that `linearity symmetry' in the
quadrature of linear ordinary second-order differential equations is reviewed,
by the use of canonical coordinates and the identification of a Wronskian-like
conserved quantity as Lie invariant. The Jacobi last multiplier associated with
two independent linearity symmetries is applied to derive the Caldirola-Kanai
Lagrangian from symmetry principles. Then the symmetry is recognized to be also
a Noether one. Finally, the study is extended to higher-order linear ordinary
differential equations, derivable or not from an action principle.
