Have I understood linear transformations correctly? I've just started self-studying Linear Algebra, and I am not quite sure I completely understand linear transformations. Here' what I've gathered for so far... 
Given a matrix $A$, we can define a linear transformation of a vector $\overline{x}$ by $T(\overline{x})= A\cdot \overline{x}$
Is this correct? It doesn't say anywhere in the textbook that we simply multiply the matrix by the vector, but the way they've written it out sure seems like matrix multiplication to me? 
Also, will the linear transformation for $A$ always be a transformation $T: \mathbb{R}^n \rightarrow \mathbb{R}^m$ where $n = columns$ of $A$ and $m = rows$ of $A$? 
Lastly, how does one figure out what matrix $A$ is behind a transformation? I know given the transformation, we can use the "standard", "basis" vectors to systematically find the columns of $A$, but what if we are given a specific set of tranformations, i.e. $T: \mathbb{R}^2 \rightarrow \mathbb{R}^3$ where  
\begin{pmatrix}2\\1\end{pmatrix} becomes \begin{pmatrix}-3\\3\\0\end{pmatrix} and \begin{pmatrix}1\\2\end{pmatrix} becomes \begin{pmatrix}3\\0\\9\end{pmatrix} What is the matrix $A$ "behind" this tranformation, or whatever you'd call it? 
 A: Yes you're correct in the two questions.
For the last question you should express the vectors of the standard basis $e_1$ and $e_2$ with the given vectors  $v_1=\begin{pmatrix}2\\1\end{pmatrix}$ and  $v_2=\begin{pmatrix}1\\2\end{pmatrix}$ which means you should find $\alpha_1,\alpha_2$ and $\beta_1,\beta_2$ such that
$$e_i=\alpha_i v_1+\beta_i v_2$$
so we get
$$Ae_i=\alpha_i Av_1+\beta_i Av_2$$
A: You don't need a basis to define a linear function. In some sense bases and matrices confuse the issue. All you need to know is that there is an underlying space $X$ with two operations defined on it: addition and scalar multiplication. Then a linear function $F$ on $X$ preserves addition and scalar multiplication, in the sense that $F(f+g)=F(f)+F(g)$ and $F(\alpha f)=\alpha F(f)$. There are ways to define such function without any mention of a basis, which means without a matrix.
A significant application where this abstraction originally arose was in the study of Heat conduction. You applied a heat source $h$ and the Response $R$ of the system was postulated to be superposition of the the responses to different heat sources. That is, if a heat source distribution function $h$ were applied and the response measured, and if another heat source $h'$ were added, then $R(h+h')=R(h)+R(h')$. Very abstract notion. And amplifying the heat source $h$ would similarly amplify the response $R$, meaning that $R(\alpha h)=\alpha R(h)$. This principle of "superposition" played a major role in the development of linear theory. While systems are not normally globally linear, there may well be a regime where they are. You add effects of sources when adding sources. You multiply the effect by the same factor by which you multiply the cause. Many of our basic laws of Physics are linear. If I add two charges into a region, the added response force fields are the superpositions of the individual force fields. The same is true of gravity. Of course, things start rearranging if the forces are too large, and that changes the system. So linearity through superposition has practical limitations, too, even when the theoretical is accurately linear.
Linear differential equations abound in Physics and Engineering. Quantum Mechanics is a linear theory, which would undoubtedly surprise many people. The first real linear theories were carried out in the context of superposition in infinite dimensional spaces, which would probably also surprise many people. The serious study of selfadjoint linear matrices was predated by the study of selfadjoint linear differential operators and their eigenfunctions in infinite dimensional spaces.
So it's probably better to think in terms of functions on a vector space where superposition holds. If you add causes $f+g$, then effects $E(f+g)=E(f)+E(g)$ are added, and if you scale causes $\alpha f$, then you scale effects $E(\alpha f)=\alpha E(f)$.
If you happen to have a linear system $F$ where all causes can be built from a finite number of causes $f_{1}, f_{2},\cdots, f_{n}$ by additions and scalings $\alpha_{1}f_{1}+\alpha_{2}f_{2}+\cdots+\alpha_{n}f_{n}$, and where the effects can be written in terms of a finite number of effects $g_{1}, g_{2},\cdots, g_{m}$ in a similar way, then you can reduce the function to a matrix of numbers, and all you need to know to reconstruct $F$ is that matrix of numbers. But the matrix of numbers should not be confused with the actual function $F$. You tabulate
$$
\begin{align}
    F(f_{1}) & = \alpha_{1,1}g_{1}+\alpha_{2,1}g_{2}+\cdots+\alpha_{m,1}g_{m} \\
    F(f_{2}) & = \alpha_{1,2}g_{1}+\alpha_{2,2}g_{2}+\cdots+\alpha_{m,2}g_{m} \\
      \vdots & = \vdots \\
    F(f_{n}) & = \alpha_{1,n}g_{1}+\alpha_{2,n}g_{2}+\cdots+\alpha_{m,n}g_{m}.    
\end{align}
$$
Because every cause $f$ is assumed to be written as the finite linear combination
$$
               f = \beta_{1}f_{1}+\beta_{2}f_{2}+\cdots+\beta_{n}f_{n}
$$
then we know the effect $F(f)$ from the tabulation $[\alpha_{j,k}]$:
$$
\begin{align}
    F(f) & =\beta_{1}F(f_{1})+\beta_{2}F(f_{2})+\cdots+\beta_{n}F(f_{n}) \\\\
       & = \beta_{1}\{ \alpha_{1,1}g_{1}+\alpha_{2,1}g_{2}+\cdots+\alpha_{m,1}g_{m}\} \\
       & + \beta_{2}\{ \alpha_{1,2}g_{1}+\alpha_{2,2}g_{2}+\cdots+\alpha_{m,2}g_{m}\} \\
       & + \cdots +\\
       & + \beta_{n}\{ \alpha_{1,n}g_{1}+\alpha_{2,n}g_{2}+\cdots+\alpha_{m,n}g_{m}\} \\\\
& = \{\alpha_{1,1}\beta_{1}+\alpha_{1,2}\beta_{2}+\cdots+\alpha_{1,n}\beta_{n}\}g_{1}\\
& + \{\alpha_{2,1}\beta_{1}+\alpha_{2,2}\beta_{2}+\cdots+\alpha_{2,n}\beta_{n}\}g_{2}\\
& + \cdots +\\
& + \{\alpha_{m,1}\beta_{1}+\alpha_{m,2}\beta_{2}+\cdots+\alpha_{m,n}\beta_{n}\}g_{m}
\end{align}
$$
You should recognize the tabular manipulation as the source of matrix multiplication.
So a linear response function $F$ is completely determined by a finite number of tabulated numbers if there are a finite number of causes and a finite number of effects from which all others can be built. And matrix multiplication becomes determined by the linearity of $F$ dictated by the principle of superposition. Conversely, every matrix can be used to defined a linear $F$ in this context, but only when there are a finite number of causes and effects. Linear functions, however, have a more general meaning, even without a basis.
