Superposition

linear-algebra

Definition

Superposition

Superposition is forming a linear combination of elements (states/inputs) in a space with addition and scalar multiplication:

$$ax+by\text{for scalars }a,b$$

A function/system ($L$) respects superposition (is linear) iff, for all $(x,y)$, and scalars $(a,b)$,

$$L(ax+by)=aL(x)+b(Ly)$$

Equivalently:

• Additivity: $L(x+y)=L(x)+L(y)$
• Homogeneity: $L(ax)=aL(x)$

Principle of Superposition

If a problem/operator $L$ is linear, then linear combinations of solutions are solutions:

$$L{x}_1=f_1,L{x}_2=f_2\Rightarrow L(a{x}_1+b{x}_2)=a{f}_1+b{f}_2.$$

In the homogeneous case $Lx=0$, any $a{x}_1+b{x}_2$ is again a solution.