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linear transformation
Let $V$ and $W$ be vector spaces over the same field $F$. A linear transformation is a function $T\colon V\to W$ such that:

$T(v+w)=T(v)+T(w)$ for all $v,w\in V$

$T(\lambda v)=\lambda T(v)$ for all $v\in V$, and $\lambda\in F$
The set of all linear maps $V\to W$ is denoted by $\operatorname{Hom}_{F}(V,W)$ or $\mathscr{L}(V,W)$.
Examples:

Let $V=\mathbb{R}^{n}$ and $W=\mathbb{R}^{m}$ and $A$ is any $m\times n$ matrix. Then the function $L_{A}:V\to W$ defined by $L_{A}(v)=Av$, the multiplication of matrix $A$ and the vector $v$ (considered as an $n\times 1$ matrix), is a linear transformation.

Let $V$ be the space of all differentiable functions over $\mathbb{R}$ and $W$ the space of all continuous functions over $\mathbb{R}$. Then $D:V\to W$ defined by $D(f)=f^{{\prime}}$, the derivative of $f$, is a linear transformation.
Properties:

$T(0)=0$.

If $S$ and $T$ are linear transformations from $V$ to $W$, and $k\in F$, then so are $S+T$ and $kT$. As a result, $\operatorname{Hom}_{F}(V,W)$ is a vector space over F.

If $G\colon W\to U$ is a linear transformations then $G\circ T\colon V\to U$ is also a linear transformation.

The image $\operatorname{Im}(T)=\{T(v)\mid v\in V\}$ is a subspace of $W$.

The inverse image $T^{{1}}(w)$ is a subspace if and only if $w=0$.

A linear transformation is injective if and only if $\operatorname{Ker}(T)=\{0\}$.

If $v\in V$ then $T^{{1}}(T(v))=v+\operatorname{Ker}(T)$.

If $w\in\operatorname{Im}(T)$ then $T(T^{{1}}(w))=\{w\}$.
Remark. A linear transformation $T:V\to W$ such that $W=V$ is called a linear operator, and a linear functional when $W=F$.
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Comments
Terminology
While in principle, the terms "map", "mapping",
"function", "transformation", etc are synonyms,
my impression is that the different terms have acquired
distinct meanings. This is more a matter of connotation than
denotation. However consistent usage that conforms to
prevalent norms should make for clearer communication.
The generic term is "mapping", or "map"  although
mapping seems to be the preferred term.
The word "function" should be reserverd for a "mapping"
whose domain and codomain are sets of "numbers" in
some general sense.
The word "transformation" should be reseverd for a
"mapping" where the domain and codomain coincide.
The basic idea is that one can compose a transformation
with itself.
The word "operator" should be reserved for "transformations"
whose domain/codomain is a set of "functions".
The word "functional" should be reserved for a mapping whose
domain is a set of "functions" and whose codomain is a set of
"numbers", in some general sense.
Re: Terminology
"Linear operator" is used in my Linear Algebra textbook (FriedbergInselSpence) so frequently (essentially half the theorems begin "Let T be a linear operator...") that I have to ask if their usage really is nonstandard.
Re: Terminology
"function" tends to have a more precise definition, in that for { (a, b) } in the graph of f, ( a ) are distinct (alternatively, f(a) has up to one value).
Re: Terminology
After looking around at texts and other references
I have to back away from my position somewhat.
The choice of terminology:
mapping vs function vs transformation vs operator
is completely unstandardized. I come from a differential
geometry background, where "mapping" is the general term
i.e. you can have mappings between manifolds, whereas
function is a "scalar field", i.e. a mapping that plunks
numbers down at points of your manifold.
So probably my original post was just expressing my
own prejudices/prefernces.