Linear transformation defined by an $m$ by $n$ matrix I came across these terms on an MIT open course on youtube:

An $m\times n$ matrix defines a linear transformation from the $\mathbb{R}^n$ vector
  space to $\mathbb{R}^m$ vector space.

Can someone please explain in simple terms what this means?
Why not a linear transformation from $\mathbb{R}^m$ to $\mathbb{R}^n$ vector space?
 A: When defining a map using a matrix, the elements of $\mathbb{R}^n$ are usually considered to be column vectors (that is, $n \times 1$ matrices). Under this convention, a matrix $A \in M_{m \times n}(\mathbb{R})$ defines a linear map $L_A \colon \mathbb{R}^n \rightarrow \mathbb{R}^m$ by left multiplication so $L_A(x) := Ax$. This is defined when $x$ is a column vector and results in a $m \times 1$ column vector.
If one would work with row vectors and right multiplication (this can be seen sometimes in old textbooks) then a matrix $A \in M_{n \times m}(\mathbb{R})$ would define a map $\mathbb{R}^n \rightarrow \mathbb{R}^m$ by $R_A(x) := xA$. This convention is more intuitive in the sense that it defines a linear map between the spaces "in the natural" order but it has other drawbacks that make it less useful. For example, when constructing a matrix that corresponds to a linear system of equations, you would have to convert the equations into columns instead of rows in order to identify the solution space with the kernel of the associated linear map.
A: I will try to explain it as simple as I can. First of all you need to recognize that a $m\times n $ matrix $A$ gives you a map from $\mathbb{R}^n$ to $\mathbb{R}^m$. This is by taking a vector $x \in \mathbb{R}^n$ and "multiplying" it in the following way:
$$ A(x)=\begin{pmatrix} a_{11} & \ldots & a_{1n}\\
\vdots & & \vdots\\
a_{m1} &\ldots & a_{mn}  \end{pmatrix}\begin{pmatrix}x_1\\\vdots\\x_n\end{pmatrix}=\begin{pmatrix}\sum_{i=1}^n x_ia_{1i}\\\vdots\\\sum_{i=1}^n x_ia_{mi} \end{pmatrix}. $$
The word "linear" comes from the relation $A(x+y)=A(x)+A(y)$ for $x,y\in \mathbb{R}^n$ and $A(\lambda x)=\lambda A(x)$ for any $\lambda \in \mathbb{R}$ (You should check this!). What does this mean? Linearity basically means that the map $A$ from   $\mathbb{R}^n$ to $\mathbb{R}^m$ transforms linear spaces into linear spaces. You can see this for example in the rule $A(\lambda x)=\lambda A(x)$. You can interpret this as following: The map $A$ sends a multiple of a vector to a multiple of it's image vector.
A: 
A linear transformation is any function $L \colon \mathbb{R}^n \to \mathbb{R}^m$ that satisfies two properties:
  
  
*
  
*$L(x + y) = L(x) + L(y)$ for all $x,y$ in $\mathbb{R}^n$; and
  
*$L(cx) = c\,L(x)$ for all $x$ in $\mathbb{R}^n$ and all real numbers $c$.
  

Meanwhile, an $m \times n$ matrix is just an $m \times n$ array of numbers.
Question: In what way does an $m \times n$ matrix (array of numbers) give rise to a linear transformation (a function satisfying those two properties)?
This was answered in complete generality by other users.  So, let me instead answer your question with an example.
Example: Suppose $A$ is the $2 \times 3$ matrix
$$A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}.$$
From $A$, we can define a function $L \colon \mathbb{R}^3 \to \mathbb{R}^2$ as follows:
$$L(x,y,z) = A \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} x + 2y + 3z \\ 4x + 5y + 6z \end{pmatrix}.$$
Since $A$ has $3$ columns, the input of $L$ has to be a vector $(x,y,z)$ in $\mathbb{R}^3$.  Since $A$ has $2$ rows, the output of $L$ has to be a vector in $\mathbb{R}^2$.  Hopefully this example illustrates the idea.
You can check that the function $L$ defined in this way is, in fact, a linear transformation, meaning that the two properties described above hold.
