Here is the single most useful consequence of linearity. If u_1 and u_2 both solve a linear, homogeneous PDE L[u] = 0, then so does every combination

The proof is one line: L[c_1 u_1 + c_2 u_2] = c_1 L[u_1] + c_2 L[u_2] = c_1\cdot 0 + c_2\cdot 0 = 0. Solutions of a linear homogeneous PDE form a vector space — you can scale them and add them and never leave the set of solutions.

Why this is the whole strategy

Superposition extends to infinite sums (with mild convergence care). That is exactly what makes the upcoming method work: we will find a stockpile of simple "building-block" solutions — products like e^{-\lambda t}\sin(kx) — and then add them in just the right amounts to match whatever starting shape we are given. The "right amounts" are Fourier coefficients.

Superposition fails for nonlinear PDEs — you cannot add two solutions of Burgers' equation and get a third. That single difference is why nonlinear PDEs are so much harder, and why this course spends most of its time in the linear world where addition is allowed.

Add two modes

Two building-block solutions on [0, \pi], u_1 = \sin x and u_2 = \sin 3x (faint), and their superposition c_1 u_1 + c_2 u_2 (bold). Dial the weights: every combination is again a valid solution, and by choosing the weights you sculpt the shape. This is a Fourier series in miniature.