Is $f: \mathbb{R^m} \rightarrow \mathbb{R^n}$ defined by $f(\vec x) \equiv 0 $ considered a linear map? If the definition for linear map is one that satisfies additivity and homogeneity, then $f$ obviously is a linear map. But I'd to confirm $f$ is not specifically excluded in the usual definitions of linear mappings.
[Math] Zero Function Linear
linear algebralinear-transformations
Best Answer
Yes, the zero function is indeed a linear map.
Any function that is additive and homogeneous is a linear map by definition, and in your case:
So $f$ is both homogeneous and additive.
In fact, as one commenter pointed out, you can define addition and multiplication with constant to the set of all linear functions.
These two definitions are well defined and map linear functions to linear functions.
Using these two operations, the set of all linear functions from $\mathbb R^m$ to $\mathbb R^n$ is a linear space (which, by the way, is isomorphic to $\mathbb R^{m\cdot n}$) and is more commonly viewed as the space of all $n\times m$ matrices) and the zero function is the zero element.