We rigorously define linearity and show that it is a constructive property: if an ODE is linear then its general solution has certain structure. That structure is codified in the structure theorem which, together with the principle of superposition, provides a blueprint for constructing general solutions of linear ODE; no such blueprint exists for nonlinear ODE. As a concrete illustration of linearity in action, we study an RC-circuit driven first by a simple harmonic and then by a linear combination of harmonics; the latter example foreshadows future discussion of Fourier analysis. Linearity is not specific to ODE but is a property of transformations acting on vector spaces. Since vector spaces are defined over number fields, we open the discussion with number fields, not just out of a sense of duty, but to dispel any lingering doubts about the validity of imaginary numbers. The subsequent explanation of vector fields and linear transformations involves many auxiliary linear algebra concepts that will be important later: linear independence, span, basis, null space, and range, to name a few. We also present matrices as representations of linear transformation which, among other things, explains the rationale behind the row-by-column matrix multiplication rule.