The term linear mapping is clear to me. It means that given a vector space $\left\{\mathfrak{u},\mathfrak{v},\dots\right\}$ over a field, $\left\{a,b,\dots\right\},$ a mapping $L$ is linear if and only if $L\left(a\mathfrak{u}+b\mathfrak{v}\right)=aL\left(\mathfrak{u}\right)+bL\left(\mathfrak{v}\right).$ If we call that mapping a transformation, saying that it is a linear transformation is uncontroversial.
In physics it is common to call a combination rotation and a translation a linear transformation. As we see, such a transformation is not a linear mapping
\begin{align*} T\left(\mathfrak{x}\right)= & \mathfrak{y}_{o}+\mathfrak{R}\mathfrak{x}=\mathfrak{y}\\ T\left(a\mathfrak{x}_{1}+b\mathfrak{x}_{2}\right)= & \mathfrak{y}_{o}+a\mathfrak{R}\mathfrak{x}_{1}+b\mathfrak{R}\mathfrak{x}_{2}\\ aT\left(\mathfrak{x}_{1}\right)+bT\left(\mathfrak{x}_{2}\right)= & \left(a+b\right)\mathfrak{y}_{o}+a\mathfrak{R}\mathfrak{x}_{1}+b\mathfrak{R}\mathfrak{x}_{2} \end{align*}
The rotational part alone does, however, qualify as a linear mapping
$$\mathfrak{R}\left(a\mathfrak{x}_{1}+b\mathfrak{x}_{2}\right)=a\mathfrak{R}\mathfrak{x}_{1}+b\mathfrak{R}\mathfrak{x}_{2}.$$
Some mathematicians, at least informally, say that a linear transformation of $\mathbb{R}^n$ leaves the origin fixed. So, my question is this: is there a generally agreed upon definition of linear transformation? If so, what is it?