9

I became very confused about linear functions after reading this question What is the difference between linear and affine function?

In the comments it says that $F(x)=2 \cdot x+4$ is not a linear function, (but an affine one). All my professors gave such examples when teaching linear functions. I am really confused now.

Should a linear function always be of the form $f(x)=t \cdot x$, where t is a constant?

I think this could help me understand better linear transformations. I think one of the reasons I did not understand them is because I had a slightly wrong definition of linear functions.

However, on Wikipedia, the definition of linear functions seems to accepts functions that also have a constant added or subtracted from the first (linear?) part.

https://en.wikipedia.org/wiki/Linear_function

https://upload.wikimedia.org/wikipedia/commons/0/0e/Linear_Function_Graph.svg

So is Wikipedia wrong on this one?

2240
  • 131
  • 6
yoyo_fun
  • 895

3 Answers3

18

Yes and no. It just depends on the context.

To repeat what the Wikipedia says:

In mathematics, the term linear function refers to two distinct but related notions:

  • In calculus and related areas, a linear function is a polynomial function of degree zero or one, or is the zero polynomial.

  • In linear algebra and functional analysis, a linear function is a linear map.

In the second case, yes, $0$ must be the fixed point of a linear map by definition.

All my professors gave such examples when teaching linear functions.

I suppose you were in the class of linear algebra.

Community
  • 1
  • 1
    So the notion of linear functions depends on which context we are talking about. This makes it hard for students to understand intuitively some concepts. – yoyo_fun Dec 11 '16 at 17:27
  • Well, usually for a mathematical concept, such as "linear functions" in your case, people might call it differently in different contexts. It is instructive to identity what they really mean by context rather than stick to a particular version of the notion. –  Dec 11 '16 at 17:34
  • Correct. It would be better if more professor would do this. – yoyo_fun Dec 11 '16 at 17:38
  • Note that in contexts where "linear" is used in the second sense (i.e. to mean a vector space (homo)morphism, general form $f(x)=ax$), one often uses the term ___affine function___ to refer to something of the form $f(x)=ax+b$. This generalizes from functions $f:\mathbb{R}\to\mathbb{R}$ to functions $f:\mathbb{R}^n\to\mathbb{R}^m$ by letting $a$ be an $m\times n$ matrix, and $b$ be a constant vector in $\mathbb{R}^m$. – Jeppe Stig Nielsen Dec 12 '16 at 10:33
4

If $L$ is a linear function, then $L(0)=L(x-x)=L(x)-L(x)=0$. So, a linear function always fix the origin.

Mr. T
  • 141
2

A function defined from $ \mathbb R \to \mathbb R$ is said to be linear if

  • $\forall x, y\; \;\;f(x+y)=f(x)+f(y)$

  • $\forall \lambda \;\;\;f(\lambda x)=\lambda x$

so, $x\mapsto ax$ satisfies these conditions while $x\mapsto ax+b$ does not and it is affine.

hamam_Abdallah
  • 62,951