# What is the difference between linear and affine function

I am a bit confused. What is the difference between a linear and affine function? Any suggestions will be appreciated

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$f(x)=2x$ is linear and affine. $f(x)=2x+3$ is affine but not linear. –  Rahul Jan 10 '13 at 18:39

A linear function fixes the origin, whereas an affine function need not do so. An affine function is the composition of a linear function with a translation, so while the linear part fixes the origin, the translation can map it somewhere else.

As an example, linear functions $\mathbb{R}^2\to\mathbb{R}^2$ preserve the vector space structure (so in particular they must fix the origin). Affine functions, however, preserve the geometry, i.e. the distance between two points. Translations preserve this just fine, even though they don't fix the origin, so there are some affine functions which aren't linear.

If you choose a basis for vector spaces $V$ and $W$, and consider functions $f\colon V\to W$, then $f$ is linear if $f(v)=Av$ for some matrix $A$ (of the appropriate size), and $f$ is affine if $f(v)=Av+b$ for some matrix $A$ and vector $b\in W$.

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Affine functions preserve the distance between two points? –  Jonas Meyer Sep 15 at 2:59
An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line. From Wikipedia –  paldepind Sep 20 at 17:46

An affine function is the composition of a linear function followed by a translation. $ax$ is linear ; $(x+b)\circ(ax)$ is affine. see Modern basic Pure mathematics : C.Sidney

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