Say if there is a matrix A:
$$\begin{bmatrix} 1 & 2 & 0 & 2 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$$
What the column space of A? : I am confused whether to exclude NON-pivot columns.
What is the dimension of column space? : The dimension of the column space or dimension of the basis of column space? Can be either 4 or 3?
What is the basis of column space? This is just the pivot columns. Is this the so called $\operatorname{Col}A$?
Best Answer
The column space is not a list of vectors so it's not clear what you mean when you ask if you should exclude non-pivot columns. The column space is the linear span of the columns. Each column (including the non-pivot columns) is contained in this space.
What you may be confusing yourself with is the column space vs. a basis for the column space. A basis is indeed a list of columns and for a reduced matrix such as the one you have a basis for the column space is given by taking exactly the pivot columns (as you have said). There are various notations for this, $\operatorname{Col}A$ is perfectly acceptable but don't be surprised if you see others.
As for the dimension of the column space, it's $3$, which is the number of elements in a basis, i.e., the number of pivot columns.