What am I understanding wrong about how matrix-norm works? I'm learning how norm works today. I think I understood how a vector norm works and now I'm trying to understand how the matrix-norm works. I can't understand why the $p=1$ is the "maximum absolute column sum of the matrix". So, here's the definition of the matrix-norm:

I wanted to use a very simple example, the same the professor gave me today
$$
 A =
 \begin{pmatrix}
  1 & 10 & 3 \\
  -5 & -1 & 0 \\
  3i  & 2  & 0
 \end{pmatrix}
$$
So, from the definition I thought that I could pick any $x \in K^3$, so I pick $x=(1, 1, 1) \Rightarrow ||x||_{p=1}=3$. Doing $||Ax||$ gets me $(1, 10, 3)$, and the norm from that is $1+10+3=14$ divided per $||x||$, $14/3$. And this makes no sense at all! I should only be able to get 3 results $(9, 13, 3)$. How do I get to these results? From the way I understood I could get unlimited results.
Many thanks in advance!

The norm is defined to be the maximum possible value you can get as you range over all $x$. For this particular $x$, you get (EDIT: $25/3$) (taking the $p = 1$ norm), but there are $x$ out there such that you get larger, so there's still only one norm of $A$. If you take the standard basis, this maximum must be attained at one of the basis elements, which is why you only need to check three numbers. So the $p = 1$ norm of your matrix is $13$.
