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es

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Most of what I have read online states, if a function has such-and-such a form then it is linear. I am hoping to form a more exact definition like, if a function has properties A and B (and perhaps C) then it is linear (or because it does not, then it is not).

I have done some research and I think these are the two properties that must hold.

For a function, f(x) where x can be a vector, to be linear the following properties must hold:

A: f(u+v)=f(u)+f(v)

B: f(c*u)=c*f(u) where c is a real scalar

So is this correct? Am I missing any properties or have I added one where I shouldn't? (On a side note, why a real scalar? Why not allow one to restrict the range?)

Also, using the above definition seems to cause some problems. For instance I always assumed the following equation was linear:

y(x)=mx+b

But property A does not hold for this function

y(u+v)=m(u+v)+b != y(u)+y(v) = m(u+v)+2b

And neither does property B

y(c*u)=m(c*u)+b != c*y(u)=c*(m*u)+c*b

One would think a line was linear and isn't y(x)=mx+b just a re-arangement of the standard form Ay+Bx+C=0?

Something seems amiss here and most likely it is me :)