What is linearity? [duplicate]
Once someone asked me the question "What is linearity?" in a proficiency exam. I went hot and cold all over. Although, I heard and even used the term linearity many many times, I had not really thought about it until that time. After a hopeless discussion, he said linearity is the satisfaction of the following conditions:
$$f(x+y)=f(x)+f(y)$$ $$f(ax)=af(x)$$
Since then I have no idea about linearity. Because there are:
- Linear equations
- Linear differential equations
- Linear algebra
- Linear programming
- Linear interpolation
- and so forth...
According to the definition even a straight line $y=mx+n$ cannot be considered linear as long as $n\neq0$. But there are so many linearities.
So, what is this term linearity?
Solution 1:
The definition of linearity depends on context.
- A linear map satisfies the conditions above.
- A linear DE means that the associated Differential operator is linear in each derivative of the unknown.
- The solutions to a linear equation are the roots of an affine linear map.
- Linear algebra deals with vector spaces and (affine) linear maps.
- Linear programming is about linear objective functions and affine constraints.
- Linear interpolation is interpolation of a function by an affine linear map.
The term affine linear used here is defined by: $f:X\to Y$ is affine linear iff there exists $a\in Y$ such that $x\mapsto f(x)-a$ is linear, i.e. $f(x) = g(x) + a$ where $g$ is linear.
Solution 2:
My answer is that linearity, in your examiner's perspective, is a canonical function between structures $X\rightarrow Y$with a commutative '$+$' and an distributive action '$\cdot$': $a\cdot(x+y)=a\cdot x + a\cdot y$. The function is such that the diagram commutes: $\require{AMScd}$ \begin{CD} A\times X\times X @>(1,f,f)>> A\times Y\times Y\\ @V S_X V V\# @VV S_Y V\\ X @>>f> Y \end{CD} That is, the function should satisfy $S_Y(1,f,f)=fS_X$. This gives the condition $S_Y((1,f,f)(a,x,y))=f(S_X(a,x,y))\Leftrightarrow S_Y(a,f(x),f(y))=f(a\cdot(x+y))\Leftrightarrow$ $a\cdot(f(x)+f(y))=a\cdot f(x) + a\cdot f(y)=f(a\cdot(x+y))$.
This seems to be possible to extend to all mathematical structures.
Linearity in your perspective perhaps referring to the lack of nonlinear variable terms.