Geometric meaning behind matrix-vector multiplication

Question

Consider a matrix $A (m , n)$ and a vector $x (n , 1)$.

I understand what the equation $Ax = b$ means ($A$ is transformation matrix and so on). I know what happens to $x$ due to this linear transformation, yet I want to consider the matrix itself so what I want to know is:

1. What is the geometrical effect imposed on the space spanned by $A$ when multiplied by $x$?

2. Is there a difference in the effect imposed on $A$ if $x$ is an eigenvector ?

Answer 1

1. I am not sure what you mean by "space spanned by $A$". $A$ maps the whole $K^n$ to a part of the $K^m$. Certain $A$ instances have a geometric interpretation, like scaling, rotation, mirroring. Or maybe you mean something like the operator norm?

2. No, the result just happens to be simple: not a linear combination of up to $n$ vectors but just a multiple of the eigenvector.