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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?
Najib Idrissi
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    Its all of the above and more; there is what might be called a family resemblence between them; but there isn't a specific technical definition that will cover them all. The general 'philosophical' idea is that they are the sum of their parts, rather than than the sum being more than their parts. – Mozibur Ullah Sep 13 '14 at 09:41
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    @MoziburUllah "sum of their parts sum being more than their parts" makes sense. You should open it with an answer. –  Sep 13 '14 at 09:51
  • There are some answers here. – Mark McClure Sep 13 '14 at 10:39
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    Did he really ask you that without more context? – Git Gud Sep 13 '14 at 10:42
  • @GitGud That was the exact question. –  Sep 13 '14 at 10:43
  • He can't seriously ask what is linearity and give the answer he gave. As the question was put, I can only guess he was testing your knowledge of the ambiguousness of the term linear or that he wanted an intuitive universal explanation (perhaps platonistic?) of what linearity is supposed to embody. By his own answer, it can't be the second option. – Git Gud Sep 13 '14 at 10:48
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    @GitGud According to the answers I am beginning to think that it was not a fair question. –  Sep 13 '14 at 11:00

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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.

Albert
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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.

AlexR
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  • As I understand there is no general definition of linearity. According to Belgi given answer is about linear functions and I think it was very misleading for me. Thank you for all answers. –  Sep 13 '14 at 11:22
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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.

Community
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Lehs
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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

Belgi
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    I wonder why linear equations [link] (http://en.wikipedia.org/wiki/Linear_equation) are not called affine equations. –  Sep 13 '14 at 09:46
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    @i.ozturk You can see them as level sets of linear maps as well: When is $ax = b$ instead of what are the roots of $ax-b$. – AlexR Sep 13 '14 at 10:15
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    In many calculus books an affine map is also called linear. An unfortunate and unnecessary term, I agree. – Git Gud Sep 13 '14 at 10:50
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    Equations ax+b=0 are sometimes called linear because they can be rewritten as L(x)=c where the function L is linear. – Did Sep 13 '14 at 10:55