What are linear combinations and linear operators in differential equations?

[JJBladester]

Homework Statement

I am in a differential equations course currently. The chapter I'm reading is "linear differerntial equations: basic theory". The words linear combination and linear operator are used.

Homework Equations

L{$$\alpha$$f(x) + $$\beta$$g(x)} = $$\alpha$$L(f(x)) + $$\beta$$L(g(x))

The Attempt at a Solution

I forgot what the quick and easy definition of "linear combination was. I seem to remember something about "closed under addition and scalar multiplication". Perhaps somebody could help me re-learn this concept. I took Calc III and Linear Algebra and passed with flying colors but time has eaten my brain.

 

[HallsofIvy]

A linear combination of, say, x₁, x₂, ..., x_n must involve only multiplication by numbers and addition or subtraction: a₁x₁+ a₂x₂+ ...+ a_nx_n where every "a" is a number. Anything more complicated than just "multiply by numbers and add", for example x² or x/y or cos(x) is NOT linear.

A linear operator is an operator that "preserves" the operations: T(ax+ by)= aT(x)+ bT(y) where a and b are numbers. Matrix multiplication of vectors, differentiation of functions and integration of functions are examples of linear operators.