linearity of the transform

Suppose you are converting a shopping list from one currency to another. You can convert each item's price separately and add them up, or add up the prices first and convert the total — you get the same answer. The Laplace transform behaves with exactly this kind of fairness: it respects addition and scaling. That property has a name — linearity — and it is what makes the transform usable by hand.

Precisely, linearity says two things at once. First, the transform of a sum is the sum of the transforms: L{f(t) + g(t)} = L{f(t)} + L{g(t)}. Second, a constant multiplier passes straight through: L{c f(t)} = c L{f(t)} for any constant c. Combined, L{a f(t) + b g(t)} = a F(s) + b G(s). This follows immediately from the definition, because the defining integral is itself linear — you can split an integral of a sum into a sum of integrals and pull constants out front.

In practice this means you never transform a complicated expression as a single lump. You break it into the simple pieces you recognize — a constant, a power, an exponential, a sine — transform each from the table, multiply by its coefficient, and add. The same logic runs in reverse for the inverse transform. Linearity is the quiet workhorse that turns a long expression into a short, mechanical lookup.