What is linearity?

Question

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?

Answer 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.

Answer 2

The question asked was a bit unclear, the answer given was to the question what is a linear function. But just a note - Indeed $$ f(x)=ax+b $$

is not linear for $b\neq0$!

It is called Affine

Answer 3

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.

Answer 4

On a "philosophical" level : a process is linear if when you double the input, you will also double the output.

The rigorous mathematical definition of linear maps, linear differential equation, etc. have already been given.

The whole idea of differential calculus is that when you zoom enough, anything looks linear at small scale : that's what a derivative is.