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?

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.
