What is a linear combination?

JJBladester

Gold Member
286
2
1. The problem statement, all variables and given/known data

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.

2. Relevant equations

L{[tex]\alpha[/tex]f(x) + [tex]\beta[/tex]g(x)} = [tex]\alpha[/tex]L(f(x)) + [tex]\beta[/tex]L(g(x))

3. 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.
 
Last edited:

HallsofIvy

Science Advisor
41,626
821
A linear combination of, say, x1, x2, ..., xn must involve only multiplication by numbers and addition or subtraction: a1x1+ a2x2+ ...+ anxn 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.
 

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