Linear function

This is a scalar-valued linear functional, so it vanishes at the origin; an affine function with a nonzero constant term is a different notion.

A real linear function on ${\mathbb{R}}^n$ has the form $x\mapsto c^{T}x$.

An affine function on a real or complex vector space has the form $x\mapsto Lx+b$, where $L$ is a linear map and $b$ is constant. Its derivative is the constant map $L$. It is linear exactly when $b=0$.

An expression of the form $a_0+{\sum }_{j=1}^m{a}_j{x}_j$, with fixed coefficients $a_j$. It contains a constant term and terms of degree one, without products of variables. It represents an affine function once the scalar domain is specified.