This article brings to light the fact that linearity is by itself a meaningful symmetry in the
senses of Lie and Noether. First, the role played by that ‘linearity symmetry’ in the quadrature of
linear second-order differential equations is revisited through the use of adapted variables and the
identification of a conserved quantity as Lie invariant. Second, the celebrated Caldirola–Kanai
Lagrangian—from which the differential equation is deducible—is shown to be naturally generated by a
Jacobi last multiplier inherited from the linearity symmetry. Then, the latter is recognised to be
also a Noether one. Finally, the study is extended to higher-order linear differential equations,
derivable or not from an action principle. Incidentally, this work can serve as an introduction to
the central question of continuous symmetries in physics and mathematics. It has the advantage of
being approachable to undergraduate students since the linearity symmetry is by its very nature
sufficient…