When we talk about a vector

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:

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 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.