Suppose I have the matrix: $$ \begin{bmatrix} 1 & 2 & 3 & 1 \\ 1 & 1 & 2 & 1 \\ 1 & 2 & 3 & 1\\ \end{bmatrix} $$
How do I figure out the number of pivot columns in its ref by using the fact that it's $3^{rd}$ and $4^{th}$ are are dependent on columns on $1$ and $2$. Please assume that the concepts of dimension and rank have not yet been introduced in the course. I wish to get an intuitive understanding.
If you're looking to determine the number of pivot columns without thinking about dimension and rank, then I would think about what happens when you row-reduce this matrix (i.e., look at the rows, not the columns). There can be no more than $3$ pivot columns, since there are only $3$ rows which could have leading entries. Since the third row is identical to the first, it will be zeroed out and will not have a leading entry. The first and second rows are not multiples of each other, though, so they should both end up with a leading entry. Hence there will be two pivot columns.