Difference between matrix mapping and matrix multiplication

Question

My textbook says that a matrix mapping is a function f: $\mathbb{R}^n\to\mathbb{R}^m$ such that $f(\vec{x}) = A\vec{x}$ where $A$ is an $m \times n$ matrix

So how is a result of a matrix mapping different from multiplying a matrix by a vector?

Is this just a way of saying that my multiplying a matrix and a vector, we are transforming the vector in some way?

Do these matrix mappings allow for some special properties of $A$?

How do definitions of four fundamental subspaces change if $A$ was a mapping vs. if it was not a mapping

Answer 1

If you consider a linear map such as $f : \mathbb{R}^{2} \to \mathbb{R}^{3}$ given by $$ f(x, y) = (2x, 3y, x + y) $$ for example, we can view this as multiplying the vector $(x, y) \in \mathbb{R}^{2}$ by the matrix $$ A = \begin{pmatrix} 2 & 0 \\ 0 & 3 \\ 1 & 1 \\ \end{pmatrix} $$ (which you can verify by determining $Ax$). In this way, we see that the matrix $A$ corresponds precisely to the linear map $f$, so that matrix multiplication by $A$ is equivalent evaluating a vector under $f$.

Answer 2

Those are completely equivalent definitions, the key fact is that a linear mapping/transformation $f(\vec{x})$, with respect to a given basis, can be expressed by a matrix multiplication $A\vec{x}$ (that is a theorem).