Linearity

[Vol]

Having difficulty with the concept of additivity in linearity. In order for a function to be linear it must be additive. f(x + y) = f(x) + f(y). y = mx + b is additive only if b = 0. Then why are all y = mx + b lines? Also, how are linear differential equations additive? Really freaked out about this Please help

 

[pka]

Having difficulty with the concept of additivity in linearity. In order for a function to be linear it must be additive. f(x + y) = f(x) + f(y). y = mx + b is additive only if b = 0. Then why are all y = mx + b lines? Also, how are linear differential equations additive?

Do not be freaked out. If you study mathematics this happens frequently.
In mathematics, the term linear function refers to two distinct but related notions.
Read this link.

 

[Vol]

OK. I read the link. But y = mx + b is a straight line for all b. But it is only additive for b = 0. f(x + y) = f(x) + f(y). Which means only y = mx + 0 is linear? Because a linear function must be additive, right? But all y = mx + b graph a straight line. Then how can they not be linear?

 

[JeffM]

I think you are missing the point. There are two completely different meanings to the term "linear function." The right to bear arms does not suggest that bears have arms rather than legs. You are trying to conflate two completely different meanings.

 

[pka]

OK. I read the link. But y = mx + b is a straight line for all b. But it is only additive for b = 0. f(x + y) = f(x) + f(y). Which means only y = mx + 0 is linear? Because a linear function must be additive, right? But all y = mx + b graph a straight line. Then how can they not be linear?

Vol, I can think right off hand where the adjective linear is used in entirely different ways: algebra, axiomatic geometry, matrices, linear algebra.
As I said before: if you do any advanced mathematics you must learn to deal with that reality.

 

[Vol]

OK. I get it. The term "linear" means different things at different places. Now I remember.
But I am having difficulty grasping why additivity and homogeneity make for linearity. Could somebody explain with examples?

 

[HallsofIvy]

In linear algebra, a function is "linear" if f(ax+ by)= af(x)+ bf(y). "Additivity" would be just "f(x+ y)= f(x)+ f(y)". "Homogeneity" also has different meanings in different areas. Do you mean just "f(ax)= af(x)"?
If so, then, yes, if f satisfies both f(x+ y)= f(x)+ f(y) and f(ax)= af(x) then f(ax+ by)= f(ax)+ f(by) (by additivity) and then = af(x)+ bf(y) (by homogeneity). Conversely if f satisfies f(ax+ by)= af(x)+ bf(y) then f(x+ y)= f(x)+ f(y) (take a= b= 1) and f(ax)= af(x) (take b= 0).