Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

share|improve this question

migrated from stats.stackexchange.com Jan 10 '13 at 18:01

This question came from our site for people interested in statistics, machine learning, data analysis, data mining, and data visualization.

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

2 Answers 2

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 $b\in W$.

share|improve this answer
Affine functions preserve the distance between two points? –  Jonas Meyer Sep 15 at 2:59

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

share|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.