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.

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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 $13$.

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I thought the $p=1$ norm was 13, from the second column? –  Clash Apr 23 '12 at 19:23
You're right, sorry, I was taking the $p = 2$ norm. Same argument, though. I'll fix my answer. –  user29743 Apr 23 '12 at 19:28
Oh jesus, I just noticed my mistake! In my, case for $x=(1,1,1)$, $14/3$ is actually completely wrong! $||Ax||$ is actually $(14, -6, 3i+2)$ and the norm is $(14+6+5)/3=25/3$. I was doing to the matrix multiplication incorrectly! Thanks for the insight with the basis elements. Using $x=(0,1,0)$ will get me $(10, -1, 2)$ and it's norm is $13$. In fact, I now see why the basis elements will always get me the sum of the absolute values of a column. –  Clash Apr 23 '12 at 19:36
Yes - note that it follows from the triangle inequality and the fact that the standard basis elements have norm one that the maximum must occur at one of the standard basis elements. –  user29743 Apr 23 '12 at 19:41
many thanks for the fast replies and answer countinghaus :) –  Clash Apr 23 '12 at 19:43