Exact formal definition for linear function

Question

Let $I_1, I_2, I_3, \dots, I_n$ be $n$ non-empty sets. And similarly $Y$ be a non-empty set.

Consider an $n$-ary function $f$ from $I_1 \times I_2 \times I_3 \times \cdots \times I_n$ to $Y$.

Then $f$ is said to a linear function if and only if the range of $f$ contains the elements of the form $a_1i_1+a_2i_2+\cdots +a_ni_n$ where $i_j \in I_j$ for all $1 \le j \le n$ and $a_i \in \mathbb{R}$ for all $1 \le i \le n$.

Is it an exact definition for the term linear function?

Answer 1

The usual definition of a linear map is the following:

Definition: Let $X$ and $Y$ be vector spaces over the same field $k$. A function $f : X \to Y$ is said to be linear if it satisfies the following two properties:

1. $f(x + y) = f(x) + f(y)$ for all $x,y\in X$, and
2. $f(ax) = af(x)$ for all $x\in X$ and $a \in k$.

Essentially, a linear map is a vector space homomorphism (i.e. a map between vector spaces which "preserves" the operations). Note that we could replace $X$ and $Y$ with modules over the same ring and get a generalized definition of linearity. I doubt that we can do much better than that and have a meaningful definition (for example, if we consider maps between groups, then the definition likely collapses to that of a "group homomorphism", which would, I think, dilute the meaning of "linear" past usefulness).

The definition of "linear" given in the question does not agree with the usual definition of "linear" as given above. For example, the function $$f : \mathbb{R} \to \mathbb{R} : x \mapsto x^3 $$ is not linear (in the usual sense), but the range of $f$ is all of $\mathbb{R}$, so it satisfies the definition of "linear" given in the question.

There is also a difficulty with the provided definition, as the $I_j$ and $Y$ are taken to be arbitrary sets. As suggested above, linear maps preserve structure, and unstructured sets lack... well... structure. Defining a map between unstructured sets as a "linear map" would, again, dilute the the meaning of "linear" beyond usefulness.

Finally, the definition provided indicates that the image of $f$ should contain all of the things which look like $$ \sum_{j=1}^{n} a_j i_j, \qquad\text{where $a_j \in \mathbb{R}$ and $i_j \in I_j$}. $$ However, these objects live in the domain of $f$ and not the codomain (they are not elements of $Y$). As such, this part of the definition is flawed, and requires some revision.