Should a linear function always fix the origin? [duplicate]

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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 $F(x)=2*x+4$ 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 $f(x)=t*x$ , 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.

So is wikipedia wrong on this one ?

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3 Answers

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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.

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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.

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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.

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