What is the difference between linear and affine function

Question

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

Answer 1

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 $b\in W$.