>>10943527I can see linearity right here in this proof:
>>10941960for example. You can think of a limit as a linear operator. Because these are all linear operations: addition and multiplication. These fractions are essentially the slope so f(a+dx)=f(a) + f'(a)dx + o(1) . As you can see with the vanishing o(1) it is simply the differential or in other words an equation of the line in the slope-intercept form. So that is the "line" in the "linear transformation".
>>10941906Yes I know now that it has everything to do with the linear approximation: it is right there in the definition of the derivative via limits.
>superposition formulaswell that's the consequence of the fact that derivative is a linear approximation. dy = mdx is the differential which is a line.
>>10941930>Linearity of the derivative has absolutely nothing to do with the fact that the derivative is related to the slope or whateveragain, thats the consequence of the fact that the derivative is a local linear approximation so you can (f+g)' = f' + g'.
>Integral is also a linear operator and integral is about areas and not about lines.Well, again you can see that the Riemann sum is a linear operation since it is just addition and multiplication. That's the reason the superposition formulas hold, not the other way around (it is not defined via the superposition requirement).
>>10941968>O[ a*f(x) + b*g(x) ] = a*O[f(x)] + b*O[g(x)].That's a general rule for all linear operators. They may be linear for different reasons but since they are linear that rule holds. But we can see why they are linear on a case per case basis. For example a matrix is a linear operation since matrix multiplication is linear. The derivative is a linear operator because it is local linear approximation due to the definition of the limit and the differential etc. Integral is a linear operator because the riemann sums are linear (multiplication + addition). etc. This is my take anyway.
