# Dimension of columnspace and nullspace

• Apr 13th 2010, 12:06 AM
Bucephalus
Dimension of columnspace and nullspace
Hi

$\displaystyle A \in \mathbb{C}^{1x3}$ and $\displaystyle A \neq 0$

What is the dimension of the range and the nullspace?

I think the columnspace has a dimension of 1 and the nullspace has a dimension of 3, but I'm not exactly sure.
Can someone confirm this for me please?

Thanks.
• Apr 13th 2010, 12:19 AM
Bucephalus
Also...
When we talk about a vector $\displaystyle x \in \mathbb{C}^3$
Are we saying it has a dimension of 3?
I always thought that but I read something once that confused me and that's why I'm here asking this relatively simple question.
for the example above the way I would have thought it would have been is this:
$\displaystyle b = Ax$ and b is in columnspace. and b has the same rows as the matrix so the columnspace in the example above has a dimension of 1.

the nullspace is all the x's in $\displaystyle b = Ax$ that make it = 0, and the x's have the same number of elements as there are columns in A, so the nullspace is dimension 3.

Is my thinking right? This is the way I used to think, but somewhere alone the way I read something and now I"m not sure. Can someone please confirm?

Thanks.