Let $S:\Bbb{R}^m\to \Bbb{R}^n$ be a linear transformation. Every textbook that I have come across states that $S$ is an $n\times m$ matrix.
It is easy to see that such a matrix satisfies the properties of such a linear transformation.
However, can $S$ be something other than a matrix? I can't think of a way of proving that $S$ can only be a matrix and nothing else.
Thanks in advance!
In general, for vector space (say over $\mathbb R$) $V$ and $W$, a linear transformation $S:V\to W$ is a function which is additive, that is $$S(x+y)=Sx+Sy$$ and homogeneous, that is $$T(\alpha x)=\alpha Tx.$$
In the particular case where $V=\mathbb R^n$ and $W=\mathbb R ^m$, where these are given the standard vector space structures, a linear transformation is a function $S:\mathbb R^n\to \mathbb R^m$ which is additive and homogeneous.
A matrix is nothing but a block of numbers. So, quite simply, a matrix is not a linear transformation, nor is a linear transformation a matrix. The reason that many textbooks blur the distinction between linear transformations and matrices is that once bases for the relevant vector spaces are given, every matrix gives rise to a unique linear transformation, and every linear transformation arises from such a matrix uniquely. In other words, there is a bijection between the set of linear transformation $\{T:V\to W\}$ and the set of matrices $M_{n,k}(\mathbb R)$ of $n\times k$ matrices with entries in $\mathbb R$, where $n$ is the dimension of $V$ and $k$ is the dimension of $W$.
So, once bases $B,B'$ are fixed, every linear transformation $T:V\to W$ is associated with a matrix $[T]_{B,B'}$ called the representing matrix of $T$ with respect to the chosen bases. Moreover, if $S:W\to U$ is another linear transformation, and $B''$ is a basis for $U$, then $[S\circ T]_{B,B''}=[S]_{B',B''}[T]_{B,B'}$. That is, composition linear transformations is represented as well by the representing matrices.
In many familiar situations there are natural choices for standard bases for the linear spaces one encounters, or, at least, there are procedures that produce explicit bases. That is why many texts don't bother to make a clear distinction between linear transformations and their representing matrices.
However, there are plenty of situations where no bases can be given explicitly, and then it is more desirable to work without coordinates.
Another reason why many textbooks prefer the 'linear transformation is a matrix' blur is that lots of things are easy to define on a matrix while they require a bit more work for general linear transformations. Examples include the trace and determinant of a linear transformation. It is possible to give coordinate-free definitions of these concepts, but this is slightly more involved than the standard matrix bases definitions.
A linear transformation is not a matrix. A matrix is a rectangular array of numbers, and a linear transformation is a mapping.
But for any linear transformation $S$ from $\mathbb{R}^m$ to $\mathbb{R}^n$, there is a unique matrix $A=A(S)$ such that for any vector $v$ in $\mathbb{R}^m$, we have $S(v)=Av$. You can write down the matrix $A(S)$ easily, column by column, if you know what the linear transformation $S$ does to the basis vectors $(1,0,\dots,0)$, $(0,1,\dots, 0)$, and so on.
Conversely, multiplication by a matrix of the right size always gives a linear transformation from $\mathbb{R}^m$ to $\mathbb{R}^n$.
