# Is an affine function with offset non-linear?

Definition: A function $$g:\Bbb R^n\ \longrightarrow\Bbb R^m$$ is affine if it is of the form $$g(x)=Mx+v$$ for some matrix $$M\in\operatorname{Mat}(n\times m,\Bbb R)$$ and vector $$v\in \Bbb R^m$$.

The matrix multiplication with $$M$$ is linear. So if there is no offset provided by $$v$$, then the affine function itself is also linear.

So let's say $$v\neq0$$. Is it then correct to say that the affine function itself (because of $$v\neq0$$) is now non-linear?

• It could be called a linear function but not in the context of linear algebra Jul 7 '19 at 13:51

Yes, an affine map $$g$$ with non-zero translation part (i.e $$v\neq 0$$) is non-linear since $$g(0)=v\neq 0$$ (a linear map sends $$0$$ to $$0$$).